Invariance Principle on the Slice

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Analysis of Boolean functions is an area at the intersection of theoretical computer science, functional analysis and probability theory, which traditionally studies Boolean functions on the Boolean cube 0, 1 n. A recent development in the area is the non-linear invariance principle of Mossel, O’Donnell and Oleszkiewiczt u [26], a vast generalization of the fundamental Berry–Esseen theorem. The Berry–Esseen theorem is a quantitative version of the Central Limit Theorem, giving bounds on the speed of convergence of a sum i Xi to the corresponding Gaussian distribution. Convergence occurs as long as none of the summands Xi is too “prominent”. The invariance principle is an analog of the Berry–Esseen theorem for low-degree polynomials.ř Given a low-degree polynomial f on n variables in which none of the variables is too prominent (technically, f has low influences), the invariance principle states that the distribution of f X1,...,Xn and f Y1,...,Yn is p q p q similar as long as each of the vectors X1,...,Xn and Y1,...,Yn consists of independent coordinates, the p q p q distributions of Xi, Yi have matching first and second moments, and the variables Xi, Yi are hypercontractive. The invariance principle came up in the context of proving a conjecture, Majority is Stablest, claiming that the majority function is the most noise stable among functions which have low influences. It is often applied in the following setting: the Xi are skewed Bernoulli variables, and the Yi are the matching normal distributions. The invariance principle allows us to analyze a function on the Boolean cube (corresponding to the Xi) by analyzing its counterpart in Gaussian space (corresponding to the Yi), in which setting it can be analyzed using geometric methods. This approach has been used to prove many results in analysis of Boolean functions (see for example [19]). The proof of the invariance principle relies on the product structure of the underlying probability spaces. The challenge of proving an invariance principle for non-product spaces seems far from trivial. Here we n prove such an invariance principle for the distribution over X1,...,Xn which is uniform over the slice rks , defined as: n n ` ˘ r s x1,...,xn 0, 1 : x1 xn k . k “ tp qPt u `¨¨¨` “ u ˆ ˙ This setting arises naturally in hardness of approximation, see e.g. [6], and in extremal combinatorics (the Erd˝os–Ko–Rado theorem and its many extensions). n Our invariance principle states that if f is a low-degree function on rks having low influences, then the distributions of f X1,...,Xn and f Y1,...,Yn are close, where X1,...,Xn is the uniform distribution n p q p q ` ˘ on rks , and Y1,...,Yn are either independent Bernoulli variables with expectation k n, or independent Gaussians with the same mean and variance. { `The˘ classical invariance principle is stated only for low-influence functions. Indeed, high-influence functions like f x1,...,xn x1 behave very differently on the Boolean cube and on Gaussian space. For the same reason,p the conditionq “ of low-influence is necessary when comparing functions on the slice and on Gaussian space. The invariance principle allows us to generalize two fundamental results to this setting: Majority is Stablest and Bourgain’s tail bound. Using Bourgain’s tail bound, we prove an analog of the Kindler–Safra theorem, which states that if a Boolean function is close to a function of constant degree, then it is close to a junta. As a corollary of our Kindler–Safra theorem, we prove a stability version of the t-intersecting Erd˝os–Ko– Rado theorem, combining the method of Friedgut [17] with calculations of Wilson [32]. Friedgut showed that n n t a t-intersecting family in rks of almost maximal size 1 ǫ k´t is close to an optimal family (a t-star) as long as λ k n 1 t 1 ζ (when k n 1 t 1 p, t´-starsq ´ are no longer optimal). We extend his result to the regimeă {k nă {p1 `t `q´1˘. { ą {p ` q ` ˘ { « {p ` q The classical invariance principle is stated for multilinear polynomials, implicitly relying on the fact that every function on 0, 1 n can be represented (uniquely) as a multilinear polynomial, and that multilinear polynomials have thet sameu mean and variance under any product distribution in which the individual factors have the same mean and variance. In particular, the classical invariance principle shows that the correct way to lift a low-degree, low-influence function from 0, 1 n to Gaussian space is via its multilinear representation. t u

1 The analogue of the collection of low degree multilinear functions on the discrete cube is given by the n collection of low degree multilinear polynomials annihilated by the operator B . Dunkl [9, 10] showed i 1 xi that every function on the slice has a unique representation as a multilinear polynomial“ B annihilated by the n ř operator B . We call a polynomial satisfying this condition a harmonic function. In a recent paper [13], i 1 xi the first author“ B showed that low-degree harmonic functions have similar mean and variance under both the uniform distributionř on the slice and the corresponding Bernoulli and Gaussian product distributions. This is a necessary ingredient in our invariance principle. Our results also apply for function on the slice that are not written in their harmonic representation. Starting with an arbitrary multilinear polynomial f, there is a unique harmonic function f˜ agreeing with f on a given slice. We show that as long as f depends on few coordinates, the two functions f and f˜ are close as functions over the Boolean cube. This implies that f behaves similarly on the slice, on the Boolean cube, and on Gaussian space. Our proof combines algebraic, geometric and analytic ideas. A coupling argument, which crucially relies on properties of harmonic functions, shows that the distribution of a low-degree, low-influence harmonic function f is approximately invariant when we move from the original slice to nearby slices. Taken together, these slices form a thin layer around the original slice, on which f has roughly the same distribution as on the original slice. The classical invariance principle implies that the distribution of f on the layer is close to its distribution on the Gaussian counterpart of the layer, which turns out to be identical to its distribution on all of Gaussian space, completing the proof. A special case of our main result can be stated as follows. Theorem 1.1. For every ǫ 0 and integer d 0 there exists τ τ ǫ, d 0 such that the following holds. Let n 1 τ, and let f be a harmonicą multilinearě polynomial of degree“ pd suchqą that with respect to the uniform ě { n measure νpn on the slice rpns , the variance of f is at most 1 and all influences of f are bounded by τ. The CDF distance between the distribution of f on the slice νpn and the distribution of f under the ` ˘ product measure µp with marginals Ber p is at most ǫ: for all σ R, p q P Pr f σ Pr f σ ǫ. | νpnr ă s´ µp r ă s| ă This result is proved in Section 5.2. Subsequent to this work, the first and third author came up with an alternative proof of Theorem 1.1 [25] which doesn’t require the influences of f to be bounded. The proof is completely different, connecting the measures µp and νpn directly without recourse to Gaussian space. While the main result of [25] subsumes the main result of this paper, we believe that both approaches have merit. Furthermore, the applications of the invariance principle appearing here are not reproduced in [25].

Paper organization An overview of our main results and methods appears in Section 2. Some preliminaries are described in Section 3. We examine harmonic multilinear polynomials in Section 4. We prove the invariance principle in Section 5. Section 6 proves Majority is Stablest, and Section 7 proves Bourgain’s tail bound, two applications of the main invariance principle. Section 8 deduces a version of the Kindler–Safra theorem from Bourgain’s tail bound. Our stability result for t-intersecting families appears in Section 9. Some open problems are described in Section 10.

2 Overview

The goal of this section is to provide an overview of the results proved in this paper and the methods used to prove them. It is organized as follows. Some necessary basic deﬁnitions appear in Subsection 2.1. The invariance principle, its proof, and some standard consequences are described in Subsection 2.2. Some applications of the invariance principle appear in Subsection 2.3: versions of Majority is stablest, Bourgain’s theorem, and the Kindler–Safra theorem for the slice. An application of the Kindler–Safra theorem to extremal combinatorics is described in Subsection 2.4. Finally, Subsection 2.5 presents results for non- harmonic multilinear polynomials.

2 2.1 Basic definitions Measures Our work involves three main probability measures, parametrized by an integer n and a probability p 0, 1 : P p q n S n S • µp is the product distribution supported on the Boolean cube 0, 1 given by µp S p| | 1 p ´| |. t u p q“ p ´ q n n • νpn is the uniform distribution on the slice r s x1,...,xn 0, 1 : x1 xn pn (we pn “ tp qPt u `¨¨¨` “ u assume pn is an integer). ` ˘ n • Gp is the Gaussian product distribution N p,...,p ,p 1 p In on Gaussian space R . pp q p ´ q q We denote by f π the L2 norm of the polynomial f with respect to the measure π. } } Harmonic polynomials As stated in the introduction, we cannot expect an invariance principle to hold for all multilinear polynomials, since for example the polynomial x1 xn pn vanishes on the slice but not on the Boolean cube or on Gaussian space. We therefore restrict`¨¨¨` our attention´ to harmonic multilinear polynomials, which are multilinear polynomials f satisfying the differential equation n f B 0. x i 1 i “ ÿ“ B (The name harmonic, whose common meaning is different, was lifted from the literature.) n Dunkl [9, 10] showed that every function on the slice rpns has a unique representation as a harmonic multilinear polynomial whose degree is at most min pn, 1 p n . This is the analog of the well-known fact ` ˘ that every function on the Boolean cube has a uniquep representatp ´ q q ion as a multilinear polynomial. One crucial property of low-degree harmonic multilinear polynomials is invariance of their L2 norm: for any p 1 2 and any harmonic multilinear polynomial f of degree d pn, ď { ď d2 f µ f G f ν 1 O . } } p “} } p “} } pn ˘ p 1 p n ˆ ˆ p ´ q ˙˙ This is proved in Filmus [13], and in fact this result (and its applications in the present work) was the main motivation for [13].

Influences The classical definition of influence for a function f on the Boolean cube goes as follows. Define i i i f r s x f xr s , where xr s results from flipping the ith coordinate of x. The ith cube-influence of f is given by p q“ p q 2 c i 2 f 1 ˆ 2 Infi f f f r s µp B f S . r s“} ´ } “ xi “ p 1 p p q › ›µp i S ›B › p ´ q ÿP This notion doesn’t make sense for functions on› the› slice, since the slice is not closed under flipping › › of a single coordinate. Instead, we consider what happens when two coordinates are swapped. Define ij ij ij f p q x f xp q , where xp q results from swapping the ith and jth coordinates of x. The i, j th slice- influencep q “ of fp is givenq by p q s E ij 2 Infij f f f p q . r s“ νpnrp ´ q s The influence of a single coordinate i is then defined as 1 n Infs f Infs f . i n ij r s“ j 1 r s ÿ“ The two definitions are related: Lemma 5.4 shows that if d O ?n then “ p q s d s Inf f Op V f Inf f . i r s“ n r s` cr s ˆ ˙

3 (The variance can be taken with respect to either the Boolean cube or the slice, due to the L2 invariance property.)

Noise stability The classical deﬁnition of noise stability for a function f on the Boolean cube goes as follows: Sc f E f x f y , ρr s“ r p q p qs where x µp and y is obtained from x by letting yi xi with probability ρ, and yi µp otherwise. The analogous„ deﬁnition on the slice is slightly more“ complicated. For a function„f on the slice,

Ss f E f x f y , ρr s“ r p q p qs n 1 1 where x νpn and y is obtained from x by doing Po ´ log random transpositions (here Po λ is „ p 2 ρ q p q a Poisson distribution with mean λ). That this deﬁnition is the correct analog can be seen through the spectral lens: Sc d d 2 Ss d d d 1 n d 2 ρ f ρ f “ µp , ρ f ρ ´ p ´ q{ f “ µpn . r s“ d } } r s“ d } } ÿ ÿ d Here f “ is the dth homogeneous part of f consisting of all monomials of degree d.

2.2 Invariance principle Our main theorem is an invariance principle for the slice.

Theorem 5.8. Let f be a harmonic multilinear polynomial of degree d such that with respect to νpn, s d K d K V f 1 and Inf f τ for all i n . Suppose that τ I´ δ and n I δ , for some constants Ip,K. r sď i r sď Pr s ď p ě p { For any C-Lipschitz functional ψ and for π Gp,µp , Pt u

E ψ f E ψ f Op Cδ . | νpnr p qs ´ π r p qs| “ p q Proof sketch. Let ψ be a Lipschitz functional and f a harmonic multilinear polynomial of unit variance, low slice-inﬂuences, and low degree d. A simple argument (mentioned above) shows that f also has low cube-inﬂuences, and this implies that k np E ψ f E ψ f Op | ´ | ?d . νk r p qs « νpnr p qs ˘ ?n ¨ ˆ ˙

x1 xn np The idea is now to apply the multidimensional invariance principle jointly to f and to S `¨¨¨` ´ , ?p 1 p n “ p ´ q deducing E ψ f 1 S σ E ψ f 1 S σ ǫ. µpr p q | |ď s“ Gpr p q | |ď s˘ n Let γp,q be the restriction of Gp to the Gaussian slice x1,...,xn R : x1 xn qn . An easy tp q P `¨¨¨` “ u argument shows that since f is harmonic, the distribution of f Gp and f γp,q is identical, and so p q p q

4 As corollaries, we bound the L´evy and CDF distances between f νpn , f µp and f Gp : p q p q p q Corollary 5.9. Let f be a harmonic multilinear polynomial of degree d such that with respect to νpn, V s f 1 and Infi f τ for all i n . There are parameters Xp,X such that for any 0 ǫ 1 2, if r s ď d X r s ďd X P r s ă ă { τ Xp´ ǫ and n Xp ǫ then the L´evy distance between f νpn and f π is at most ǫ, for π Gp,µp . Inď other words, forě all σ,{ p q p q Pt u

Pr f σ ǫ ǫ Pr f σ Pr f σ ǫ ǫ. νpnr ď ´ s´ ď π r ď sď νpnr ď ` s`

Corollary 5.11. Let f be a harmonic multilinear polynomial of degree d such that with respect to νpn, V s f 1 and Infi f τ for all i n . There are parameters Yp, Y such that for any 0 ǫ 1 2, if r s “ d Yd r s ď d YdP r s ă ă { τ Ypd ´ ǫ and n Ypd ǫ then the CDF distance between f νpn and f π is at most ǫ, for ď p q ě p q { p q p q π Gp,µp . In other words, for all σ, Pt u Pr f σ Pr f σ ǫ. | νpnr ď s´ π r ď s| ď

The proofs of these corollaries closely follows the proof of the analogous results in [26].

2.3 Applications As applications to our invariance principle, we prove analogues of three classical results in analysis of Boolean functions: Majority is stablest; Bourgain’s theorem; and the Kindler–Safra theorem:

n s Theorem 6.3. Let f : rpns 0, 1 have expectation µ and satisfy Infi f τ for all i n . For any 0 ρ 1, we have Ñ r s r s ď P r s ă ă ` ˘ 1 s log log α 1 1 S f Γρ µ Op,ρ Oρ , where α min τ, , ρr sď p q` log 1 ` n “ p n q ˆ α ˙ ˆ ˙ 1 where Γρ µ is the probability that two ρ-correlated Gaussians be at most Φ´ µ (here Φ is the CDF of a standardp Gaussian).q p q

n s k Theorem 7.2. Fix k 2. Let f : rpns 1 satisfy Infi f ď τ for all i n . For some constants 1 ě C Ñ t˘Cu r s ď P r s Wp,k, C, if τ W ´ V f and n Wp,k V f then ď p,k r s ě` ˘ { r s V k 2 f f ą Ω r s . } } “ ?k ˆ ˙ n k 2 Theorem 8.5. Fix the parameter k 2. Let f : rpns 1 satisfy f ą ǫ. There exists a function n ě Ñ t˘ u } } “ h: r s 1 of degree k depending on Ok,p 1 coordinates (that is, invariant under permutations of all pn Ñ t˘ u p q` ˘ other coordinates) such that ` ˘ 2 1 C 1 f h Op,k ǫ { , } ´ } “ ` n1 C ˆ { ˙ for some constant C. The proof of Theorem 6.3 closely follows its proof in [26]. The proofs of the other two theorems closely follows analogous proofs in [21].

2.4 t-Intersecting families As an application of our Kindler–Safra theorem, we prove a stability result for t-intersecting families. First, a few deﬁnitions:

5 n • A t-intersecting family F r s is one in which A B t for any A, B F. Ď k | X |ě P n • A t-star is a family of the form` ˘A r s : A J , where J t. t P k Ě u | |“ n • A t, 1 -Frankl family is a family of the` form˘ A r s : A J t 1 , where J t 2. p q t P k | X |ě ` u | |“ ` Ahlswede and Khachatrian [1, 3] proved that if n ` t ˘ 1 k t 1 and F is an intersecting family, n t ą p ` qp ´ ` q then F k´t , and furthermore equality holds if and only if F is a t-star. They also proved that when n |t |1 ď k ´ t 1 the same upper bound holds, but now equality holds for both t-stars and t, 1 -Frankl families.“ p ` qp` ´ ˘` q p q A corresponding stability result was proved by Friedgut [17]:

1 n Theorem 9.2. Let t 1, k t, λ, ζ 0, and λn k t 1 ζ n. Suppose F rks is a t-intersecting ě n tě n ą ă ă p ` ´ q Ď family of measure F k´t ǫ k . Then there exists a family G which is a t-star such that | |“ ´ ´ ` ˘ ` ˘ ` ˘ F△G | n | Ot,λ,ζ ǫ . k “ p q Friedgut’s theorem requires k n to be bounded` ˘ away from 1 t 1 . Using the Kindler–Safra theorem on the slice rather than the Kindler–Safra{ theorem on the Boolean{p ` cubeq (which is what Friedgut uses), we can do away with this limitation: Theorem 9.3. Let t 2, k t 1 and n t 1 k t 1 r, where r 0. Suppose that k n λ for ě ně ` “ p ` qp ´ ` q` nąt n { ě some λ 0. Suppose F rks is a t-intersecting family of measure F k´t ǫ k . Then there exists a family Gąwhich is a t-starĎ or a t, 1 -Frankl family such that | |“ ´ ´ ` ˘p q ` ˘ ` ˘ 1 C F△G k { 1 C 1 | | Ot,λ max , 1 ǫ { , n “ r ` n1 C k ˜ ˜ˆ ˙ ¸ { ¸ for some constant C. ` ˘ C 1 Furthermore, there is a constant At,λ such that ǫ At,λ min r k, 1 ` implies that G is a t-star. ď p { q Our proof closely follows the argument of Friedgut [17], transplanting it from the setting of the Boolean cube to the setting of the slice, using calculations of Wilson [32] in the latter setting. The argument involves certain subtelties peculiar to the slice.

2.5 Non-harmonic functions All results we have described so far apply only to harmonic multilinear polynomials. We mentioned that some n of these results trivially don’t hold for some non-harmonic multilinear polynomials: for example, i 1 xi np doesn’t exhibit invariance. This counterexample, however, is a function depending on all coordinates.“ ´ In contrast, we can show that some sort of invariance does apply for general multilinear polynomialsř that depend on a small number of coordinates: Theorem 4.3. Let f be a multilinear polynomial depending on d variables, and let f˜be the unique harmonic n multilinear polynomial agreeing with f on r s , where d pn n 2. For π µp, Gp we have pn ď ď { Pt u ` ˘ d22d f f˜ 2 O f 2 . } ´ }π “ p 1 p n } }π ˆ p ´ q ˙ Proof sketch. Direct calculation (appearing in Lemma 4.2) shows that if ω is a Fourier character than

d2 ω ω˜ 2 ω ω˜ 2 O , } ´ }µp “} ´ }Gp “ p 1 p n ˆ p ´ q ˙

6 whereω ˜ is deﬁned analogously to f˜. We can assume without loss of generality that f depends only on the variables in d 1,...,d . Since ˜ ˆ r s“t u f S d f S ω˜S, “ Ďr s p q ř d2 d22d f f˜ 2 2d fˆ S 2O O f 2 , } ´ }π ď p q p 1 p n “ p 1 p n } }π S d ˆ ˙ ˆ ˙ ÿĎr s p ´ q p ´ q using the Cauchy–Schwartz inequality. The idea of the proof is to prove a similar results for Fourier characters (Lemma 4.2) for individual Fourier characters, and then to invoke the Cauchy–Schwartz inequality. As a consequence, if we have a multilinear polynomial f depending on a small number of variables, its harmonic projection f˜ (deﬁned as in the theorem) has a similar expectation, L2 norm, variance and noise stability (Corollary 4.4). This implies, for example, that our Majority is stablest theorem is tight: the harmonic projection of the majority of a small number of indices serves as the tight example.

3 Preliminaries

Notation The notation 1E is the characteristic function of the event E. Expectation, variance and covariance are denoted by E, V and Cov, respectively. The sign function is denoted sgn. The notation n denotes n r s the set 1,...,n . The slice rks consists of all subsets of n of cardinality k. We often identify subsets of n witht their characteristicu vectors in 0, 1 n. r s r s The notation Bin n,p denotes` ˘ a binomialt u distribution with n trials and success probability p. The notation Po λ denotesp aq Poisson distribution with expectation λ. The notation N µ, Σ2 denotes a normal distributionp withq mean µ and covariance matrix Σ2. For a scalar p, we use p to denotep aq constant p vector (of appropriate dimension which is clear from context) and In to denote the n n identity matrix. 2 ˆ For a probability distribution π, f f π Eπ f is the L2 norm of f with respect to π. Note that f E f . } }“} } “ r s 1 a } }The“ notationr| |s ab denotes the falling factorial function: ab a a 1 a b 1 . Asymptotic notation (O and the like) will always denote“ non-negativep ´ q¨¨¨p expressions.´ ` q When the expression can be positive or negative,p¨q we use the notation O . The underlying limit is always n . If the hidden ˘ p¨q Ñ8 constant depends on variables V , we use the notation OV . A C-Lipschitz functional is a function ψ : R R satisfyingp¨q ψ x ψ y C x y , which implies that for functions f,g on the same domain: Ñ | p q´ p q| ď | ´ | Lemma 3.1. For every C-Lipschitz functional ψ and functions f,g on the same domain,

E ψ f E ψ g C f g . | r p qs ´ r p qs| ď } ´ } Probability distributions Our argument will involve several diﬀerent probability distributions on Rn (where n will always be clear from context):

n S n S • µp is the product distribution supported on 0, 1 given by µp S p| | 1 p ´| |. t u p q“ p ´ q n • νk is the uniform distribution on the slice rks .

• Gp is the Gaussian product distribution N ` p,...,p˘ ,p 1 p In . pp q p ´ q q n 1 n 1 p 1 p • γp,q N q,...,q , Σ , where Σi,j ´n p 1 p δ i j ´n pn ´1 q δ i j for 1 i, j n. “ pp q q “ p ´ q p “ q´ ´ p ‰ q ď ď As is well-known, the distribution γp,q results from conditioning Gp on the sum being qn.

Lemma 3.2. Let X1,...,Xn Gp. The distribution of X1,...,Xn conditioned on X1 Xn qn p q„ p q `¨¨¨` “ is γp,q.

7 Proof. Let S X1 Xn, and consider the multivariate Gaussian distribution X1,...,Xn,S , whose “ `¨¨¨` p q In 1 distribution is easily calculated to be N p pn ,p 1 p . Let Y1,...,Yn be the distribution of p p ´ q 11 n q p q ˆ ˙ X1,...,Xn,S conditioned on S qn,` which is˘ well-known to be multivariate Gaussian. Using well-known p q “ 1 formulas, the mean of this distribution is p 1n´ qn pn q (as can be derived directly), and its 1 ` p ´ q “ V 1 covariance matrix is p 1 p In 1n´ 11 . The diagonal elements are Yi p 1 p 1 n and the p ´ qp ´ q 1 r s “ p ´ qp ´ q oﬀ-diagonal ones are Cov Yi, Yj p 1 p . p q“ p ´ qp´ n q We can also go in the other direction.

p 1 p Lemma 3.3. Let X1,...,Xn γp,q, let Y N p q, p ´ q , and let Yi Xi Y . Then Y1,...,Yn Gp. p q„ „ p ´ n q “ ` p q„ Proof. As is well-known, Y1,...,Yn is a multivariate Gaussian, and it is easy to see that its mean is p. We have V Yi V Xi V Y p 1 p and Cov Yi, Yj Cov Xi,Xj V Y 0. The lemma follows. r s“ r s` r s“ p ´ q p q“ p q` r s“ The distributions µp and νk are very close for events depending on o ?n coordinates. p q Lemma 3.4. Let A be an event depending on J coordinates, where J 2 n. Then ď J 2 νpn A µp A µp A . | p q´ p q| ď 4p 1 p n p q p ´ q Proof. The triangle inequality shows that we can assume that A is the event x1 xℓ 0, xℓ 1,...,xJ ℓ J ℓ “¨¨¨“ “ ` “ 1 for some ℓ. Let k pn. Clearly µp A 1 p p ´ , whereas “ p q “ p ´ q ℓ J ℓ n k k ´ νpn A p ´ q . p q“ nJ We have nJ 1 J 1 1 J 1 J 2 1 1 ´ 1 `¨¨¨`p ´ q 1 . nJ “ ´ n ¨ ¨ ¨ ´ n ě ´ n ě ´ 2n ˆ ˙ ˆ ˙ Therefore ℓ J ℓ 2 n k k ´ J νpn A p ´ q µp A 1 , p qď nJ 1 J 2 2n ď p q ` n p ´ {p qq ˆ ˙ 1 using 1 x 1 2x, which is valid for x 1 2. Similarly,´ ď ` ď {

ℓ J ℓ 2 2 2 2 2 n k k ´ ℓ J ℓ J J J p ´ q 1 p ´ q 1 max , 1 . n k ℓkℓ ě ´ 2 1 p n ´ 2pn ě ´ 2 1 p n 2pn ě ´ 4p 1 p n p ´ q p ´ q ˆ p ´ q ˙ p ´ q Therefore n k ℓkℓ 1 J 2 4p 1 p n J 2 νpn A p ´ q p ´ {p p ´ q qq µp A 1 . p qě nℓ “ p q ´ 4p 1 p n ˆ p ´ q ˙ This completes the proof.

3.1 Harmonic multilinear polynomials n Rn Our argument involves extending a function over a slice rks to a function on , just as in the classical invariance principle, a function on 0, 1 n is extended to Rn by writing it as a multilinear polynomial. In our case, the correct way of extending at functionu over a slice to`Rn˘is by interpreting it as a harmonic multilinear polynomial. Our presentation follows [13], where the proofs of various results claimed in this section can be found. The basis in Deﬁnition 3.7 below also appears in earlier work of Srinivasan [30], who constructed it and showed that it is orthogonal with respect to all exchangeable measures.

8 2 R f Definition 3.5. Let f x1,...,xn be a formal polynomial. We say that f is multilinear if Bx2 0 for P r s B i “ all i n . We say that f is harmonic if Pr s n f B 0. x i 1 i “ ÿ“ B The somewhat mysterious condition of harmonicity arises naturally from the representation theory of the Johnson association scheme. Just as any function on the Boolean cube 0, 1 n can be represented uniquely as a multilinear polynomial (up to an affine transformation, this is just thet Fourier–Walshu expansion), every n function on the slice rks can be represented uniquely as a harmonic multilinear polynomial, using the identification ` ˘ n n n r s x1,...,xn 0, 1 : xi k . k “ tp qPt u “ u ˆ ˙ i 1 ÿ“ n Lemma 3.6 ([13, Theorem 4.1]). Every real-valued function f on the slice rks can be represented uniquely as a harmonic multilinear polynomial of degree at most min k,n k . p ´ q ` ˘ There is a non-canonical Fourier expansion defined for harmonic multilinear polynomials.

Deﬁnition 3.7. Let A a1,...,ad and B b1,...,bd be two sequences of some common length d of distinct elements of n .“ We p say that qA B if:“ p q r s ă (a) A and B are disjoint.

(b) B is monotone increasing: b1 bd. ă¨¨¨ă (c) ai bi for all i d . ă Pr s A sequence B b1,...,bd is a top set if A B for some sequence A. The collection of all top sets of “ p q ă length d is denoted Bn,d, and the collection of all top sets is denoted Bn. If A a1,...,ad and B b1,...,bd satisfy A B, deﬁne “ p q “ p q ă d

χA,B xai xbi . “ i 1p ´ q ź“ For a top set B, deﬁne χB χA,B. “ A B ÿă Finally, deﬁne χd χ 2,4,...,2d . “ t u Lemma 3.8 ([13, Theorem 3.1,Theorem 3.2]). Let π be any exchangeable distribution on Rn (that is, π is invariant under permutation of the coordinates). The collection Bn forms an orthogonal basis for all harmonic multilinear polynomials in R x1,...,xn (with respect to π), and r s n 2 2 bi 2 i 1 χB π cB χ B π, where cB ´ p ´ q , } } “ } | |} “ 2 i 1 ˆ ˙ ź“ and χB π denotes the norm of χB with respect to π. In} particular,} if f is a harmonic multilinear polynomial then E f is the same under all exchangeable measures. r s Lemma 3.3 and Lemma 3.8 put together have the surprising consequence that harmonic multilinear functions have exactly the same distribution under Gp and γp,q.

9 Lemma 3.9. Let f be a harmonic multilinear polynomial. The random variables f Gp ,f γp,q are identically distributed. p q p q

p 1 p Proof. According to Lemma 3.3, if x1,...,xn γp,q, y N q p, p ´ q and yi xi y, then p q „ „ p ´ n q “ ` y1, ,yn Gp. The lemma follows since yi yj xi xj and a harmonic multilinear polynomial p ¨ ¨ ¨ q „ ´ “ ´ can be expressed as a function of the diﬀerences xi xj for all i, j. ´ Lemma 3.8 allows us to compare the norms of a harmonic multilinear function under various distributions.

n Corollary 3.10. Let π1, π2 be two exchangeable distributions on R , and let f be a harmonic multilinear 2 2 2 polynomial of degree d. If for some ǫ 0 and all 0 e d it holds that 1 ǫ χe π1 χe π2 1 ǫ χe π1 , then also 1 ǫ f 2 f 2 1 ěǫ f 2 . ď ď p ´ q} } ď} } ď p ` q} } p ´ q} }π1 ď} }π2 ď p ` q} }π1 The following lemma records the norms of basis elements for the distributions considered in this paper. Lemma 3.11. For all d we have

2 2 d χd χd 2p 1 p , } }µp “} }Gp “ p p ´ qq d d 2 2 d pn 1 p n d d χd 2 p q pp ´ q q 2p 1 p 1 O . } }νpn “ n2d “ p p ´ qq ˘ p 1 p n ˆ ˆ p ´ q ˙˙ 2 2 Proof. The exact formulas for χd and χd are taken from [13, Theorem 4.1]. Since x1,...,xn are } }µp } }νpn G χ 2 E x x 2 d p p d independent under p, we have d Gp 1 2 2 1 . } } “ rp2 ´ q s “ p p ´ qq It remains to prove the estimate for χd . The proof of [13, Theorem 4.1] shows that } }νpn 2 2 O d O d d d 1 p q 1 p q 2 d pn 1 p n d ´ pn ´ 1 p n χd 2 p q pp ´ q q 2p 1 p p ´ q . νpn n2d ´ ¯´O d2 ¯ } } “ “ p p ´ qq 1 p q ´ n ´ ¯ It follows that 2 2 2 2 2 χd ν d d d d } } pn 1 O 1 O . 2p 1 p d “ ˘ pn ` 1 p n ` n “ ˘ p 1 p n p p ´ qq ˆ p ´ q ˙ ˆ p ´ q ˙ Lemma 3.11 and Corollary 3.10 imply an L2 invariance principle for low degree harmonic multilinear polynomials. Corollary 3.12. Suppose f is a harmonic multilinear polynomial of degree d on n variables. For any p 1 2 ď { such that d pn and any π µp, Gp we have ď Pt u d2 f ν f π 1 O . } } pn “} } ˘ p 1 p n ˆ ˆ p ´ q ˙˙ 3.2 Analysis of functions

n n We consider functions on three diﬀerent kinds of domains: the Boolean cube 0, 1 , the slice rks , and n t u Gaussian space R . We can view a multilinear polynomial in R x1,...,xn as a function over each of these domains in the natural way. r s ` ˘ For each of these domains, we proceed to deﬁne certain notions and state some basic results. The material for the Boolean cube and Gaussian space is standard, and can be found for example in [28].

10 Functions on the Boolean cube The Boolean cube is analyzed using the measure µp for an appropriate n p. The Fourier characters ωS and Fourier expansion of a function f : 0, 1 R are given by t u Ñ xi p ωS x1,...,xn ´ , f fˆ S ωS. p q“ “ p q i S p 1 p S n źP p ´ q Ďrÿ s k ˆ a We define f “ S k f S ωS, and so a multilinear polynomial f of degree d can be decomposed as 0 “d | |“ p q 0 d f f “ f “ . Since the Fourier characters are orthogonal, the parts f “ ,...,f “ are orthogonal. In the “ `¨¨¨` ř k k k 0 k future it will be convenient to separate f into f f ď f ą for an appropriate k, where f ď f “ f “ k k 1 d “ ` “ `¨¨¨` and f ą f “ ` f “ . “ i `¨¨¨ì i Define f r s x f xr s , where xr s results from flipping the ith coordinate of x. The ith cube-influence is given by p q“ p q 2 c i 2 f 1 2 Inf f f f r s B fˆ S . i r s“} ´ } “ x “ p 1 p p q › i › S i ›B › p ´ q ÿQ c n c › › The total influence of f is Inf f i 1 Infi f , and it satisfies the Poincaréinequality r s“ “ r s › › V řf p 1 p Infc f deg f V f . r sď p ´ q r s ď p q r s The noise operator Tρ is defined by deg f i i Tρf ρ f “ . “ i 0 ÿ“ The noise stability of f at ρ is deg f Sc i i 2 ρ f f,Tρf ρ f “ . r s“x y“ i 0 } } ÿ“ The noise operator (and so noise stability) can also be defined non-spectrally. We have Tρf x E f y , p qp q“ r p qs where y is obtained from x by letting yi xi with probability ρ, and yi µp otherwise. “ „

n Functions on the slice The slice rks is analyzed using the measure νk. The corresponding notion of Fourier expansion was described in Section 3.1. A harmonic multilinear polynomial f of degree d can 0 `d ˘ k be decomposed as f f “ f “ , where f “ contains the homogeneous degree k part. The parts 0 d “ `¨¨¨` f “ ,...,f “ are orthogonal. s E ij 2 ij ij ij The i, j th influence of a function f is Infij f f f p q , where f p q x f xp q , and xp q p q r s “ rp ´ q s p q “ p q s is obtained from x by swapping the ith and jth coordinates. We define the ith influence by Infi f 1 n s s n s r s “ n j 1 Infij f , and the total influence by Inf f i 1 Infi f . The total influence satisfies the Poincaré inequality“ r s r s“ “ r s ř V f Infs f ř deg f V f . r sď r s ď p q r s For a proof, see for example [13, Lemma 5.6]. The noise operator Hρ is defined by

deg f d 1 d 1 n d Hρf ρ p ´p ´ q{ qf “ . “ d 0 ÿ“ The noise stability of f at ρ is

deg f Ss d 1 d 1 n d 2 ρ f f,Hρf ρ p ´p ´ q{ q f “ . r s“x y“ d 0 } } ÿ“ The noise operator (and so noise stability) can also be deﬁned non-spectrally. We have Hρf x n 1 1 p qp q “ E f y , where y is obtained from x by taking Po ´ log random transpositions. r p qs p 2 ρ q

11 n Functions on Gaussian space Gaussian space is R under a measure Gp for an appropriate p. In this paper, we mostly consider functions on Rn given by multilinear polynomials, and these can be expanded in terms of the ωS. General functions can be expanded in terms of Hermite functions. Every square- k k k integrable function can be written as f k 0 f “ , where f “ satisfies f “ αx1 p,...,αxn p k k “ ě p ` ` q “ α f “ x1 p,...,xn p . p ` ` q ř The distributions µp and Gp have the same first two moments, and this implies that Eµ f EG f and p r s“ p r s f µ f G for every multilinear polynomial f. The Ornstein–Uhlenbeck operator Uρ is defined just like } } p “} } p Tρ is defined for the cube. Noise stability is defined just like in the case of the cube, and we use the same notation Sc for it. The noise operator (and so noise stability) can also be defined non-spectrally. We have Uρf x p qp q “ E f y , where y 1 ρ p ρx 1 ρ2 N 0,p 1 p . We can also define noise stability as Sc f r p qs “ p ´ q ` ` ´ p p ´ qq ρr s “ p 1 p ρp 1 p E f x f y , where x, y N p,p , a p ´ q p ´ q . r p q p qs p q„ p q ρp 1 p p 1 p ˆ p ´ q p ´ q ˙ ` ˘ k Homogeneous parts For a function f, we have defined f “ in three different ways, depending on the domain. When f is a harmonic multilinear polynomial, all three definitions coincide. Indeed, any harmonic B multilinear polynomial is a linear combination of functions of the form χA,B. We show that χA,B χ“| | “ A,B under all three definitions. Let A a1,...,ak and B b1,...,bk. Since χA,B is homogeneous of degree k “ k “ as a polynomial, we see that χA,B χ“ over the slice. Also, “ A,B k k 2 xai p xbi p χA,B p 1 p { ´ ´ . “ p p ´ qq i 1 ˜ p 1 p ´ p 1 p ¸ ź“ p ´ q p ´ q a a Opening the product into a sum of terms, we can identify each term with a basis function ωS for some S of k size k. This shows that χA,B χA,B“ over the cube. Finally, since χA,B is harmonic, in order to show that k “ k χA,B χA,B“ in Gaussian space, it suffices to show that χA,B αx α χA,B x , which is true since χA,B is homogeneous“ of degree k as a polynomial. p q“ p q

Degrees The following results state several ways in which degree for functions on the slice behaves as expected. First, we show that degree is subadditive. Lemma 3.13. Let f,g be harmonic multilinear polynomials, and let h be the unique harmonic multilinear n polynomial agreeing with fg on the slice r s . Then deg h deg f deg g. k ď ` Proof. We can assume that deg f deg g` k˘, since otherwise the result is trivial. ` ď i Let Ei be the operator mapping a function φ on the slice to the function φ“ on the slice. That is, we take the harmonic multilinear representation of φ, extract the i’th homogeneous part, and interpret the result as d a function on the slice. Also, let E d i 0 Ei. A function φ on the slice has degree at most d if and only ď “ “ if it is in the range of E d. ď ř Qiu and Zhan [29] (see also Tanaka [31]) show that fg is in the range of E deg f E deg g, where is ď ˝ ď ˝ the Hadamard product. The operators Ei are the primitive idempotents of the Johnson association scheme (see, for example, [4, §3.2]). Since the Johnson association scheme is Q-polynomial (cometric), the range of E deg f E deg g equals the range of E deg f deg g, and so deg fg deg f deg g. ď ˝ ď ď ` ď ` As a corollary, we show that “harmonic projection” doesn’t increase the degree. Corollary 3.14. Let f be a multilinear polynomial, and let g be the unique harmonic multilinear polynomial n agreeing with f on the slice r s . Then deg g deg f. k ď ` ˘

12 Proof. When f x1, one checks that g is given by the linear polynomial “ 1 n k g x x . n 1 i n “ i 1p ´ q` ÿ“ The corollary now follows from Lemma 3.13 and from the easy observation deg αF βG max deg F, deg G . p ` qď p q

An immediate corollary is that degree is substitution-monotone.

Corollary 3.15. Let f be a harmonic multilinear polynomial, let g x1,...,xn p q “ f x1,...,xn 1,b for b 0, 1 , and let h be the unique harmonic multilinear polynomial agreeing with p ´ nq P t u g on the slice r s . Then deg h deg f. k ď ` ˘ Noise operators We have considered two noise operators, Hρ and Tρ Uρ. Both can be applied syntactically on all multilinear polynomials. The following result shows that both“ operators behave the same from the point of view of Lipschitz functions. Lemma 3.16. Let f be a multilinear polynomial of degree at most n 2. For δ 1 2 and any C-Lipschitz functional ψ, and with respect to any exchangeable measure, { ă { 2 Cδ´ E ψ H1 δf E ψ U1 δf O f . | r p ´ qs ´ r p ´ qs| “ n } } ˆ ˙ Proof. Let ρ 1 δ. Lemma 3.1 shows that “ ´ n 2 { 2 2 2 2 d 1 d 1 n d 2 d 2 E ψ Hρf E ψ Uρf C Hρf Uρf C ρ p ´p ´ q{ q ρ f “ . | r p qs ´ r p s| ď } ´ } “ d 0p ´ q } } ÿ“ x d 1 d 1 n d d d 1 Let R x ρ . Then ρ p ´p ´ q{ q ρ p n´ q R1 x for some x d 1 d 1 n , d . For such x, p qx “ d 1 d 1 n ´ “d 2 p´ p qq P r p ´ p ´ q{ q s R1 x ρ log ρ ρ p ´p ´ q{ q 2δ ρ { 2δ , using δ 1 2 and d n 2. Therefore p q“ p´ qď p qď p q ă { ď { d 1 d 1 n d d d 1 d 2 ρ p ´p ´ q{ q ρ 2δ p ´ qρ { . ´ ď n 2 3 d d d 2 3 The expansion x 1 x d8 0 2 x implies that d d 1 ρ { 2ρ 1 ?ρ . Since 1 ?ρ 1 ?1 δ δ 2, we conclude{p that´ q “ “ p ´ q ď {p ´ q ´ “ ´ ´ ě { ř ` ˘ 2 d 1 d 1 n d 32δ´ ρ p ´p ´ q{ q ρ . ´ ď n The lemma follows.

4 On harmonicity

Let f be a function on the Boolean cube 0, 1 n, and let f˜ be the unique harmonic function agreeing with f n ˜ t u n on the slice rpns . We call f the harmonic projection of f with respect to the slice rpns . In this section we prove Theorem 4.3, which shows that when f depends on 1 ǫ log n variables, it is close to its harmonic ` ˘ p ´ q ` ˘ projection under the measure µp. Together with Corollary 3.12, this allows us to deduce properties of f on the slice given properties of f on the Boolean cube, an idea formalized in Corollary 4.4. We start by examining single monomials. Lemma 4.1. Let m be a monomial of degree d, and let f be the unique harmonic multilinear polynomial n agreeing with m on r s (where d k n 2). Then deg f d and the coeﬃcient cm of m in f is k ď ď { “ ` ˘ n 2d 1 d cm ´ ` 1 O . “ n d 1 “ ´ n ´ ` ˆ ˙

13 Proof. Without loss of generality we can assume that m xn d 1 xn. Let B n d 1,...,n . Recall “ ´ ` ¨ ¨ ¨ “t ´ ` u that the basis element χB is equal to

χB xa1 xn d 1 xad xn . “ p ´ ´ ` q¨¨¨p ´ q a1 ad n d ‰¨¨¨‰ÿPr ´ s n Let f be the unique harmonic multilinear polynomial agreeing with m on rks . Corollary 3.14 shows that deg f d. The coeﬃcient fˆ B of χB in the Fourier expansion of f is given by the formula fˆ B ď2 p q ` ˘ p q “ f,χB χB . Since deg f d, it is not hard to check that in the Fourier expansion of f, the monomial m x y{} } ď only appears in χB . Therefore the coeﬃcient cm of m in f is

d d f,χB cm 1 n d x 2y, “ p´ q p ´ q χB } } d 2 since there are n d summands in the deﬁnition of χB. The value of χB is given by Lemma 3.8 and Lemma 3.11: p ´ q } } d d 2 n d 1 n d n 2d 2 d k n k χB ´ ` ´ ´ ` 2 p ´ q . } } “ 2 2 ¨ ¨ ¨ 2 n2d ˆ ˙ˆ ˙ ˆ ˙ n We proceed to compute f,χB . Let S rks . If f S χB S 0 then B S, which happens with probability d d x y P p q pd q‰ Ď k n . The number of non-zero terms (each equal to 1 ) is the number of choices of a1,...,ad S, namely { d d d ` ˘ d d p´ q R n k . Therefore f,χB 1 k n k n , and so p ´ q x y“p´ q p ´ q { d d 2d d k n k n cm n d p ´ q “ p ´ q ¨ nd ¨ n d 1 n d 2 n 2d 2 2 n 2d 1 kd n k d p ´ ` qp ´ q ¨¨¨p ´ ` q p ´ ` q p ´ q n d d n d d p ´ q p ´ q “ n d 1 n d 2 n 2d 2 2 n 2d 1 p ´ ` qp ´ q ¨¨¨p ´ ` q p ´ ` q n 2d 1 ´ ` . “ n d 1 ´ ` Finally, since cm 0 and deg f d, we can conclude that deg f d. ‰ ď “ As a consequence, we obtain a result on Fourier characters on the cube.

Lemma 4.2. Let ω ωS be a Fourier character with respect to the measure µp of degree d, and let ω˜ be the “ n unique harmonic multilinear polynomial agreeing with ω on rnps (where d np n 2). For π µp, Gp we have ď ď { P t u d2` ˘ ω ω˜ 2 O . } ´ }π “ p 1 p n ˆ p ´ q ˙ Proof. Recall that 1 ω x p . p p d 2 i “ 1 { i Sp ´ q p p ´ qq źP Lemma 4.1 shows that c d ω˜ xi η, c 1 O , “ p 1 p d 2 ` “ ´ n { i S ˆ ˙ p p ´ qq źP where η involves other monomials. In fact, sinceω ˜ is harmonic, it is invariant under shifting all the variables by p, and so ω˜ cω η1, “ ` where η1 involves other characters. Due to orthogonality of characters we have d ω˜ ω 2 ω˜ 2 2c 1 ω 2 ω˜ 2 2c 1 ω˜ 2 1 O . } ´ }π “} }π ´ p ´ q} }π “} }π ´ p ´ q“} }π ´ ` n ˆ ˙

14 Sinceω ˜ is harmonic, Corollary 3.12 allows us to estimate ω˜ 2 given ω˜ 2 , which we proceed to estimate: } }π } }νpn

d t d t 2 1 d k n k ´ 2t 2 d t ω˜ p ´ q 1 p p p ´ q νpn p p d t nd } } “ 1 t 0 p ´ q p p ´ qq ÿ“ ˆ ˙ d 2 1 d t d t 2t 2 d t d p 1 p ´ 1 p p p ´ q 1 O “ p 1 p d t p ´ q p ´ q ˘ p 1 p n t 0 ˆ ˙ ˆ ˆ ˙˙ p p ´ qq ÿ“ p ´ q d2 1 O . “ ˘ p 1 p n ˆ p ´ q ˙ Corollary 3.12 shows that the same estimate holds even with respect to π, and so

d2 d2 ω˜ ω 2 ω˜ 2 1 O O . } ´ }π “} }π ´ ` p 1 p n “ p 1 p n ˆ p ´ q ˙ ˆ p ´ q ˙ We can now conclude that a multilinear polynomial depending on a small number of variables is close to its harmonic projection. Theorem 4.3. Let f be a multilinear polynomial depending on d variables, and let f˜ be the unique harmonic n multilinear polynomial agreeing with f on r s , where d pn n 2. For π µp, Gp we have pn ď ď { Pt u ` ˘ d22d f f˜ 2 O f 2 . } ´ }π “ p 1 p n } }π ˆ p ´ q ˙ Proof. We can assume without loss of generality that f depends on the ﬁrst d coordinates. Express f as a ˆ ˜ ˆ linear combination of characters: f S d f S ωS. Clearly f S d f S ω˜S, whereω ˜S is the unique “ Ďr s p q “ Ďr s p q ω n function agreeing with S on rpns . Lemmař 4.2 together with the Cauchy–Schwartzř inequality shows that ` ˘ d2 d22d f f˜ 2 2d fˆ S 2O O f 2 . } ´ }π ď p q p 1 p n “ p 1 p n } }π S d ˆ ˙ ˆ ˙ ÿĎr s p ´ q p ´ q This completes the proof. Combining Theorem 4.3 with Corollary 3.12, we show how to deduce properties of f on the slice given its properties on the cube. Corollary 4.4. Let f be a multilinear polynomial depending on d variables, and let f˜ be the unique harmonic n 2 2 multilinear polynomial agreeing with f on r s , where d pn n 2. Suppose that f f 1. For pn ď ď { } }µp “} }Gp ď π µp, Gp we have: Pt u ` ˘ 2 d2d{ 1. Eπ f Eν f˜ Op . | r s´ pn r s| “ p ?n q 2 d2d{ 2. f˜ ν 1 Op . } } pn “ ˘ p ?n q 2 d2d{ 3. V f˜ ν V f π Op . r s pn “ r s ˘ p ?n q 2 c c d2d{ 4. For all ρ 0, 1 , S f˜ ν S f π Op . Pr s ρr s pn “ ρr s ˘ p ?n q 2 ℓ ℓ d2d{ 5. For all ℓ d, f˜“ ν f “ µ Op . ď } } pn “} } p ˘ p ?n q

15 Proof. Throughout the proof, we are using Corollary 3.12 to convert information on f˜ with respect to π to information on f˜ with respect to νpn. All calculations below are with respect to π. For the ﬁrst item, note that

d 2 d2 { E f E f˜ f f˜ 1 f f˜ 2 Op . | r s´ r s|ď} ´ } ď} ´ } “ ?n ˆ ˙ The second item follows from the triangle inequality

f f f˜ f˜ f f f˜ . } }´} ´ }ď} }ď} }`} ´ }

For the third item, notice first that E f f 1 f 2 1. The item now follows from the previous two. | r s|ď } } ď } } “ Sc The fourth item follows from the fact that ρ is 1-Lipschitz, which in turn follows from the fact that c 2 S f T ρf and that T ρ is a contraction. ρr s“} ? } ? For the fifth item, assume that f depends on the first d variables, and write f S d cSωS. We have “ Ďr s

ℓ ℓ ℓ ℓ ř f˜“ cS ω˜S “ f cS ω˜S “ . “ p q “ “ ` p q S d S ℓ ÿĎr s | ÿ|ą Ą 2 ℓ 2 2 d Lemma 4.2 shows that for S ℓ, ω˜S “ ωS ω˜S Op . Therefore | |ą }p q } ď} ´ } “ p n q 2 d ℓ ℓ 2 d 2 d d 2 f˜“ f 2 c Op Op . } ´ “ } ď S n “ n S ℓ ˆ ˙ ˆ ˙ | ÿ|ą Ą The ﬁfth item now follows from the triangle inequality and the second item.

5 Invariance principle

In the sequel, we assume that parameters p 0, 1 2 and n such that pn is an integer are given. The assumption p 1 2 is without loss of generality.P p { s ď { We will use big O notation in the following way: f Op g if for all n N p , it holds that f C p g, “ p q ě p q ď p q where N p , C p are continuous in p. In particular, for any choice of pL,pH satisfying 0 pL pH 1, p q p q ă ď ă if p pL,pH then f O g . Stated diﬀerently, as long as λ p 1 λ, we have a uniform estimate P r s “ p q ď ď ´ f Oλ g . Similarly, all constants depending on p (they will be of the form Ap for various letters A) depend continuously“ p q on p.

Proof sketch Let ψ be a Lipschitz functional and f a harmonic multilinear polynomial of unit variance, low slice-inﬂuences, and low degree d. A simple argument shows that f also has low cube-inﬂuences, and this implies that k np E ψ f E ψ f Op | ´ | ?d . νk r p qs « νpnr p qs ˘ ?n ¨ ˆ ˙ x1 xn np The idea is now to apply the multidimensional invariance principle jointly to f and to S `¨¨¨` ´ , ?p 1 p n “ p ´ q deducing E ψ f 1 S σ E ψ f 1 S σ ǫ. µpr p q | |ď s“ Gpr p q | |ď s˘ An application of Lemma 3.9 shows that

E ψ f 1 S σ Pr S σ E ψ f . Gpr p q | |ď s“ Gp r| |ď s Gpr p qs

16 Similarly, E ψ f 1 S σ Pr S σ E ψ f Op σ?d . µpr p q | |ď s“ µp r| |ď spµpr p qs ˘ p qq

Since PrG S σ Prµ S σ Θp σ , we can conclude that p r| |ď s« p r| |ď s“ p q ǫ E ψ f E ψ f Op σ?d . νpnr p qs « Gpr p qs ˘ ` σ ´ ¯ By choosing σ appropriately, we balance the two errors and obtain our invariance principle. For minor technical reasons, instead of using 1 S σ we actually use a Lipschitz function supported on S σ. | |ď | |ď Main theorems Our main theorem is Theorem 5.8, proved in Section 5.1 on page 20. This is an invariance principle for low-degree, low-influence functions and Lipschitz functionals, comparing the uniform measure on the slice νpn to the measure µp on the Boolean cube and to the Gaussian measure Gp. Some corollaries appear in Section 5.2 on page 21. Corollary 5.9 gives a bound on the Lévy distance between the distributions f νpn and f Gp for low-degree, low-influences functions. Corollary 5.11 gives p q p q a bound on the CDF distance between the distributions f νpn and f Gp for low-degree, low-influences functions. Corollary 5.12 extends the invariance principle top functioq ns ofp arbitraryq degree to which a small amount of noise has been applied.

5.1 Main argument

We start by showing that from the point of view of L2 quantities, distributions similar to µp behave similarly. Deﬁnition 5.1. Let p 0, 1 . A parameter q 0, 1 is p-like if p q p 1 p n. A distribution is p-like if it is one of theP following: p q µ ,ν , G , whereP p q qis p-like. | ´ | ď p ´ q{ q qn q a Lemma 5.2. Let f be a harmonic multilinear polynomial of degree d ?n, and let π1, π2 be two p-like distributions. Then ď 2 2 f f 1 Op d ?n . } }π1 “} }π2 ˘ { 2 s ij 2 c f 2 The same holds if we replace f with Inf f f` f ` or Inf˘˘ f B . ij p q i xi } } r s“} ´ } r s“} B } Furthermore, there is a constant Sp such that if d Sp?n then for all p-like distributions π1, π2, ď 1 f 2 } }π1 2. 2 ď f 2 ď } }π2 D Proof. Let αD q 2q 1 q , where D d. An easy calculation shows that αD1 q 2 1 2q D 2q 1 D 1 p q “ p p ´ qq ď p q“ p ´ q p p ´ q ´ , and in particular α1 q O DαD q q 1 q . It follows that for p-like q, αD q αD p 1 qq | Dp q| “ p p q{ p ´ qq p q “ p qp ˘ Op D ?n . Lemma 3.11 thus shows that for π µq,νqn, Gq and all D d, p { qq Pt u ď 2 2 2 D D D D χD αD q 1 Op 2p 1 p 1 Op . } }π “ p q ˘ n “ p p ´ qq ˘ ?n ` n ˆ ˆ ˙˙ ˆ ˆ ˙˙ Since D d and d ?n implies d2 n d ?n, we conclude that ď ď { ď {

2 D d χD 2p 1 p 1 Op . } }π “ p p ´ qq ˘ ?n ˆ ˆ ˙˙ The lemma now follows from Corollary 3.10.

We single out polynomials whose degree satisﬁes d Sp?n. ď Deﬁnition 5.3. A polynomial has low degree if its degree is at most Sp?n, where Sp is the constant in Lemma 5.2.

17 We can bound the cube-inﬂuence of a harmonic multilinear polynomial in terms of its slice-inﬂuence. Lemma 5.4. Let f be a harmonic multilinear polynomial of low degree d, and let π be a p-like distribution. For all i n , with respect to π: Pr s

c d s Inf f Op V f Inf f . i r sď n r s` i r s ˆ ˙

Proof. We will show that for the product measure π µp it holds that “ 2d 2n Infc f V f Infs f i r sď p 1 p n d r s` p 1 p n d i r s p ´ qp ´ q p ´ qp ´ q which will imply the statement of the lemma by Lemma 5.2. s The idea is to come up with an explicit expression for Infi f . Let j i. For S not containing i, j we have r s ‰ ij ij ij ij ωSp q ωS, ωSp q i ωS j , ωSp q j ωS i , ωSp q i,j ωS i,j . “ Yt u “ Yt u Yt u “ Yt u Yt u “ Yt u Therefore s ij 2 ˆ ˆ 2 Infij f f f p q f S i f S j . r s“} ´ } “ i,j Sp p Yt uq´ p Yt uqq ÿR On the other hand, we have 1 1 p 1 p Infc f fˆ S 2 fˆ S 2 n S fˆ S i 2. i n d n d p ´ q r s“ S i p q ď S i p q p ´ | |q “ j i i,j S p Yt uq ÿQ ´ ÿQ ´ ÿ‰ ÿR The L2 triangle inequality shows that fˆ S i 2 2fˆ S j 2 2 fˆ S i fˆ S j 2, and so p Yt uq ď p Yt uq ` p p Yt uq ´ p Yt uqq 2 2 p 1 p Infc f fˆ S j 2 Infs f i n d n d ij p ´ q r sď j i i,j S p Yt uq ` j i r s ´ ÿ‰ ÿR ´ ÿ‰ 2p 1 p 2n p ´ q Infc f Infs f n d j n d i ď j i r s` r s ´ ÿ‰ ´ 2d 2n V f Infs f , ď n d r s` n d i r s ´ ´ using the Poincar´einequality. Rearranging, we obtain the statement of the lemma. Using Lemma 5.4, we can show that the behavior of a low degree function isn’t too sensitive to the value of q in νqn. Lemma 5.5. Let f be a harmonic multilinear polynomial of low degree d, and let ℓ be an integer such that νℓ is p-like. For every C-Lipschitz functional ψ,

E E d V ψ f x ψ f x Op C f νpn . | x νℓr p p qqs ´ x νℓ`1r p p qqs| “ n r s „ „ ˜ c ¸ Proof. Let q ℓ n, which is p-like. For i n , let Xi, Y i be the distribution obtained by choosing i “ {n i i Pi r s p q i i i i a random X r szt u and setting Y X i . Note that f X f Y f f r s X . Since P ℓ “ Yt u p q´ p q “ p ´ qp q Prν xi 0 1 q, we have ℓ r “ s“ ´` ˘

i i 2 1 c d s E f X f Y 1 q ´ Inf f ν Op V f ν Inf f ν , rp p q´ p qq s ď p ´ q i r s ℓ “ n r s pn ` i r s pn ˆ ˙ using Lemma 5.4 and Lemma 5.2.

18 n n Consider now the distribution X, Y supported on rℓs ℓr s1 obtained by taking X νℓ and choosing p q ˆ ` „ Y X uniformly among the n ℓ choices; note that Y νℓ 1. Since X, Y is a uniform mixture of the distributionsĄ Xi, Y i , we deduce´ ` „˘ `` ˘ p q p q

2 d 1 s E f X f Y Op V f ν Inf f ν rp p q´ p qq sď n r s pn ` n r s pn ˆ ˙ d Op V f ν , ď n r s pn ˆ ˙ using the Poincar´einequality Infs f d V f (see Section 3.2). The lemma now follows along the lines of Lemma 3.1. r s ď r s We now apply a variant of the invariance principle for Lipschitz functionals due to Isaksson and Mossel.

Proposition 5.6 ([18, Theorem 3.4]). Let Q1,...,Qk be n-variate multilinear polynomials of degree at most V c d such that with respect to µp, Fi 1 and Infj Fi τ for all i k and j n . For any C-Lipschitz k r s ď r s ď P r s P r s functional Ψ: R R (i.e., a function satisfying Ψ x Ψ y C x y 2), Ñ | p q´ p q| ď } ´ } E E d 1 6 Ψ Q1,...,Qk Ψ Q1,...,Qk Ok Cρpτ { , | µpr p qs ´ Gpr p qs| “ p q for some (explicit) constant ρp 1. ě Lemma 5.7. Denote n i 1 xi np S “ ´ . “ p 1 p n ř p ´ q Let f be a harmonic multilinear polynomial of lowa degree d 1 such that with respect to µp, E f 0, V s ě d d r s “ f 1 and Infi f τ for all i n . Suppose that τ Rp´ and n Rp, for some constant Rp. For anyr sC ď-Lipschitz functionalr s ď ψ suchP that r sψ 0 0 and B-Lipschitzď functionalě φ (where B 1) satisfying φ 1, p q “ ě } }8 ď 1 12 E E d 2 d { ψ f φ S ψ f φ S COp ?Bρp{ τ . | µpr p q p qs ´ Gpr p q p qs| “ ` n ˜ ˆ ˙ ¸ s c The condition Inf f τ for all i n can be replaced by the condition Inf f µ τ for all i n . i r sď Pr s i r s p ď Pr s Proof. For M to be chosen later, deﬁne

M if ψ x M, ´ p qď´ ψ˜ x ψ x if M ψ x M, p q“ $ p q ´ ď p qď &’M if M ψ x . ď p q ’ It is not hard to check that ψ˜ is also C-Lipschitz.% We are going to apply Proposition 5.6 with Q1 f, Q2 S p 1 p n, and Ψ y1,y2 ψ˜ y1 φ y2 . V c “ “ { p ´ q p q “ p q p q With respect to µp, Q2 1 and Infi Q2 1 p 1 p n for all i n . Lemma 5.4 shows that c d r s “ r s “ {p p ´ q q a P r s d Inf f Op τ , and so the cube-inﬂuences of Q1,Q2 are bounded by Op τ . Since i r s“ p n ` q p ` n q

Ψ y1,y2 Ψ z1,z2 Ψ y1,y2 Ψ y1,z2 Ψ y1,z2 Ψ z1,z2 | p q´ p q| ď | p q´ p q| ` | p q´ p q| ď MB y2 z2 C y1 z1 , | ´ |` | ´ | we see that Ψ is MB C -Lipschitz. Therefore p ` q 1 6 E ˜ E ˜ d d { ψ f φ S ψ f φ S Op MB C ρp τ . | µpr p q p qs ´ Gpr p q p qs| “ p ` q ` n ˜ ˆ ˙ ¸

19 Next, we want to replace ψ˜ with ψ. For π µp, Gp we have Pt u ˜ E ψ f φ S E ψ f φ S E ψ f φ S 1 ψ f M | π r p q p qs ´ π r p q p qs| ď π r| p q| | p q| | p q|ě sď 2 2 C 2 C C E f 1 f M C E f . π r| | | |ě { sď M π r| | sď M Therefore 1 6 2 E E d d { C ψ f φ S ψ f φ S Op MB C ρp τ . | µpr p q p qs ´ Gpr p q p qs| “ p ` q ` n ` M ˜ ˆ ˙ ¸ 1 6 d d d d 1 6 Choosing M C Bρ τ { completes the proof. The conditions on τ,n guarantee that ρ τ { “ { p ` n pp ` n q ď 1, and so B 1 allowsb us to obtain the stated error bound. ě ` ˘ In order to ﬁnish the proof, we combine Lemma 5.7 with Lemma 5.5.

Theorem 5.8. Let f be a harmonic multilinear polynomial of degree d such that with respect to νpn, V f 1 s d K d K r sď and Inf f τ for all i n . Suppose that τ I´ δ and n I δ , for some constants Ip,K. For any i r sď Pr s ď p ě p { C-Lipschitz functional ψ and for π Gp,µp , Pt u

E ψ f E ψ f Op Cδ . | νpnr p qs ´ π r p qs| “ p q

s c The condition Inf f τ for all i n can be replaced by the condition Inf f µ τ for all i n . i r sď Pr s i r s p ď Pr s Proof. We prove the theorem for π Gp. The version for µp then follows from the classical invariance principle, using Lemma 5.4. “ Replacing f with f E f (recall that the expectation of f is the same with respect to both µp and π) doesn’t change the variance´ r ands influences of f, so we can assume without loss of generality that E f 0. E E r s“ Similarly, we can replace ψ with ψ ψ 0 without affecting the quantity νpn ψ f µp ψ f , and so we can assume without loss of generality´ thatp q ψ 0 0. r p qs ´ r p qs For a parameter σ 1 to be chosen later,p defineq“ a function φ supported on σ, σ by ď r´ s 1 x σ if σ x 0, φ x ` { ´ ď ď p q“ 1 x σ if 0 x σ. # ´ { ď ď Note that φ 1 and that φ is 1 σ -Lipschitz. Lemma 5.7 (together with Lemma 5.2) shows that } }8 “ p { q 1 12 E E 1 2 d 2 d { ψ f φ S ψ f φ S COp σ´ { ρp{ τ | µpr p q p qs ´ Gpr p q p qs| “ ` n ˜ ˆ ˙ ¸ 1 2 d 2 1 12 1 24 COp σ´ { ρ { τ { n´ { , “ p p p ` qq d d assuming τ R´ and n R (the condition on n implies that d is low degree). ď p ě p Let α be the distribution of x1 xn under Gp. Lemma 3.2 and Lemma 3.9 show that `¨¨¨` q np E ψ f φ S E E ψ f φ ´ Gp q α γp,q ?p 1 p n r p q p qs “ „ r p qs p p ´ q q q np E ψ“ f E φ ´ ‰ E ψ f E φ S . Gp q α ?p 1 p n Gp Gp “ r p qs „ r p p ´ q qs “ r p qs r p qs

20 Similarly, Lemma 5.5 shows that

k np k np d Pr S ´ φ ´ k np Op C µp ?p 1 p n ?p 1 p n ď r “ p ´ q s p p ´ q q| ´ | ˜ n¸ k np σ?p 1 p n c | ´ |ďÿ p ´ q E φ S Op Cσ?d . ď µpr p qs p q Therefore

E ψ f E φ S E ψ f E φ S | Gpr p qs Gpr p qs ´ νnpr p q µpr p qs| ď 1 2 d 2 1 12 1 24 E Op Cσ´ { ρp{ τ { n´ { Op Cσ?d φ S . p p ` qq ` p µpr p qsq Proposition 5.6 shows that

1 2 1 6 E φ S E φ S Op σ´ { n´ { . | Gpr p qs ´ µpr p qs| “ p q

Moreover, EG ψ f C EG f Op C . It follows that p r p qs ď p r| |s “ p q E E E 1 2 d 2 1 12 1 24 E φ S ψ f ψ f Op Cσ´ { ρp{ τ { n´ { Op Cσ?d φ S . µpr p qs| Gpr p qs ´ νnpr p qs| ď p p ` qq ` p µpr p qsq

9 It is not hard to check that EG φ S Θp σ , and so for n Apσ´ we have Eµ φ S Θp σ , implying p r p qs “ p q ě p r p qs “ p q E E 3 2 d 2 1 12 1 24 ψ f ψ f Op Cσ´ { ρp{ τ { n´ { Cσ?d . | Gpr p qs ´ νnpr p qs| ď p p ` q` q

d 5 1 12 1 24 2 5 1 5 Choosing σ ρp{ τ { n´ { { d { , we obtain “ p ` q { E E d 5 1 30 1 60 3 10 ψ f ψ f Op Cρp{ τ { n´ { d { . | Gpr p qs ´ νnpr p qs| ď p p ` q q

β 9 1 γd It is not hard to check that if d Bpn and n Mp then n Apσ´ , and that if τ,n´ ρp´ then σ 1; these are the conditions necessaryď for ourě estimate to hold.ě In fact, for an appropriateď choice of ď γpd β γp γ, the condition n ρp implies the condition d Bpn , and furthermore allows us to estimate 1ě60 3 10 1 70 ě ď n´ { d { Op n´ { (say), and to control the other error term similarly. This completes the proof of the theorem.“ p q

5.2 Corollaries Theorem 5.8 allows us to bound the L´evy distance between the distribution of a low degree polynomial with respect to νpn and the distribution of the same polynomial with respect to Gp or µp. This is the analog of [26, Theorem 3.19(28)].

Corollary 5.9. Let f be a harmonic multilinear polynomial of degree d such that with respect to νpn, V f 1 s r sďd X and Infi f τ for all i n . There are parameters Xp,X such that for any 0 ǫ 1 2, if τ Xp´ ǫ r sd ď X P r s ă ă { ď and n Xp ǫ then the L´evy distance between f νpn and f π is at most ǫ, for π Gp,µp . In other words,ě for all{ σ, p q p q P t u Pr f σ ǫ ǫ Pr f σ Pr f σ ǫ ǫ. νpnr ď ´ s´ ď π r ď sď νpnr ď ` s`

21 Proof. Given σ and ǫ, deﬁne a function ψ by

0 if x σ, x σ ď ψ x ´ if σ x σ ǫ, p q“ $ ǫ ď ď ` &’1 if x σ ǫ. ě ` ’ d K d k Note that ψ is 1 ǫ -Lipschitz. Theorem 5.8 shows% that if τ I´ δ and n I δ , p { q ď p ě p {

Pr f σ Pr f σ ǫ E ψ f E ψ f Op δ ǫ . π r ď s´ νpnr ď ` sď π r p qs ´ νpnr p qs “ p { q

2 We can similarly get a bound in the other direction. To complete the proof, choose δ cpǫ for an appropriate “ cp. Using the Carbery–Wright theorem, we can bound the actual CDF distance. This is the analog of [26, Theorem 3.19(30)]. V Proposition 5.10 (Carbery–Wright). Let f be a polynomial of degree at most d such that f Gp 1. Then for all ǫ 0 and all x, r s “ ą 1 d Pr f x ǫ O dǫ { . Gp r| ´ |ď s“ p q

Corollary 5.11. Let f be a harmonic multilinear polynomial of degree d such that with respect to νpn, V s f 1 and Infi f τ for all i n . There are parameters Yp, Y such that for any 0 ǫ 1 2, r s “ d Yd r s ď d YdP r s ă ă { if τ Ypd ´ ǫ and n Ypd ǫ then the CDF distance between f νpn and f π is at most ǫ, for ď p q ě p q { p q p q π Gp,µp . In other words, for all σ, Pt u Pr f σ Pr f σ ǫ. | νpnr ď s´ π r ď s| ď

Proof. It is enough to prove the corollary for π Gp, the other case following from the corresponding result “ d X in the classical setting. Corollary 5.9 and the Carbery–Wright theorem show that for τ Xp´ η and d X ď n Xp η we have ě { 1 d Pr f σ Pr f σ η η Pr f σ Op dη { . νpnr ď sď Gp r ď ` s` ď Gp r ď s` p q d We can similarly obtain a bound from the other direction. To complete the proof, choose η cp ǫ d for “ p { q an appropriate cp. All bounds we have considered so far apply only to low degree functions. We can get around this restriction by applying a small amount of noise to the functions before applying the invariance principle itself. This is the analog of [26, Theorem 3.20]. Even though the natural noise operator to apply on the slice is Hρ, from the point of view of applications it is more natural to use Uρ (which we apply syntactically). Lemma 3.16 shows that the diﬀerence between the two noise operators is small.

Corollary 5.12. Let f be a harmonic multilinear polynomial such that with respect to νpn, V f 1 and s r s ď Infi f τ for all i n . There is a parameter Zp such that for any 0 ǫ 1 2 and 0 δ 1 2, if r Zsp ďδ PZp r δ s ă ă { ă ă { τ ǫ { and n 1 ǫ { then for π Gp,µp , ď ě { Pt u

E ψ U1 δf E ψ U1 δf Op Cǫ . | νpnr p ´ qs ´ π r p ´ qs| “ p q

d d Proof. Let g U1 δf. Let d be a low degree to be decided, and split g gď gą . With respect to νpn, d 2 “ ´ 2t t 2 2d “ ` d K gą t d 1 δ f “ 1 δ . On the other hand, Theorem 5.8 shows that if τ Ip´ ǫ and } }d “K ą p ´ q } } ď p ´ q ď n Ip ǫ then ě { ř d d E ψ gď E ψ gď Op Cǫ . | νpnr p qs ´ π r p qs| “ p q

22 d d Since g gď gą , Lemma 3.1 and Lemma 5.2 show that as long as the degree d is low, } ´ }“} } d δd E ψ g E ψ g Op Cǫ C 1 δ Op Cǫ Ce´ . | νpnr p qs ´ π r p qs| “ p ` p ´ q q“ p ` q

Choosing d log 1 ǫ δ, the resulting error is Op Cǫ . This degree is low if log 1 ǫ δ Sp?n, a condition which is implied“ byp { theq{ stated condition on n. p q p { q{ ď

6 Majority is stablest

Recall Borell’s theorem. Rn n Sc Theorem 6.1 (Borell [5]). Let f : 0, 1 have expectation µ with respect to N 0, 1 . Then ρ f Ñ r s 1 p q r s ď Γρ µ , where Γρ µ is the probability that two ρ-correlated Gaussians be at most Φ´ µ . p q p q p q Borell’s theorem remains true if we replace the standard Gaussian with Gp. Indeed, given a function f, define a new function g by g x f p 1 p x p . If x N 0, 1 then p 1 p x p Gp, and so E E p Sqc “ p Spc ´ q ` q „ p q 2p ´ q ` „ f Gp g N 0,1 . Similarly, ρ f Gp ρ g N 0,1 . Indeed, if y ρx 1 ρ N 0, 1 then r s “ r s p q r s a“ r s p q “ ` á p q p 1 p y p p ρ p 1 p x 1 ρ2 N 0,p 1 ap p ´ q ` “ ` p ´ q ` ´ p p ´ qq a 1 ρap ρ p 1 ap x p 1 ρ2 N 0,p 1 p . “ p ´ q ` p p ´ q ` q` ´ p p ´ qq Therefore Borell’s theorem for f and Gp follows froma the theorem forag and N 0, 1 . Majority is stablest states that a similar bound essentially holds for all lowp influenceq functions on the slice. This result was originally proved using the invariance principle in [26]. An alternative inductive proof appears in [7]. 1 It is known (see for example [26]) that the bound Φ´ µ is achieved by threshold functions. Corollary 3.12 together with Lemma 3.16 shows that threshold functionsp q achieve the bound also on the slice. Indeed, take E Sc ˜ a threshold function f on d variables such that with respect to µp, f µ and ρ f Γρ µ ǫ. Let f be n r s“ r sěc p qć the restriction of f to the slice r s . Corollary 3.12 shows that E f˜ µ on 1 and S f˜ S f on 1 . pn r s“ ˘ p q ρr s“ ρr s˘ p q Ss ˜ Sc E ˜ Sc Lemma 3.16 shows that ρ f ` ρ˘ f on 1 . Therefore for large n, f µ and ρ f Γρ µ 2ǫ. Our proof of majority isr stablests“ r s˘ closelyp followsq the proof of [26, Threorems« 4.4] presentedr sě inp [28,q´ §11.7]. We need an auxiliary result on Γρ.

Proposition 6.2 ([26, Lemma B.6]). For each ρ, the function Γρ deﬁned in Theorem 6.1 is 2-Lipschitz.

n s Theorem 6.3. Let f : rpns 0, 1 have expectation µ and satisfy Infi f τ for all i n . For any 0 ρ 1, we have Ñ r s r s ď P r s ă ă ` ˘ 1 s log log α 1 1 S f Γρ µ Op,ρ Oρ , where α min τ, . ρr sď p q` log 1 ` n “ p n q ˆ α ˙ ˆ ˙ s c The condition Inf f τ for all i n can be replaced by the condition Inf f µ τ for all i n . i r sď Pr s i r s p ď Pr s n Proof. We identify f with the unique harmonic multilinear polynomial agreeing with it on rpns . For a n parameter 0 δ 1 2 to be chosen later, let g H1 δf. Note that the range of g on r s is included in ă ă { “ ´ pn ` ˘ 0, 1 as well, since H1 δ is an averaging operator. We have r s ´ ` ˘ pn Ss Ss Ss Ss d 1 d 1 n 2d 1 d 1 n d 2 ρ f ρ g ρ f ρ 1 δ 2 f ρ p ´p ´ q{ q 1 1 δ p ´p ´ q{ q f “ . p ´ q r s´ r s“ r s´ r s“ d 0 p ´ p ´ q q} } ÿ“ Since d n 2, we have ď { d 1 d 1 n 2d 1 d 1 n d 2 2d d 2 ρ p ´p ´ q{ q 1 1 δ p ´p ´ q{ q ρ { 1 1 δ 2δdρ { . p ´ p ´ q qď p ´ p ´ q qď

23 2 d d 2 2 The expansion x 1 x dx shows that dρ { ?ρ 1 ?ρ , and so {p ´ q “ d ď {p ´ q ř ?ρ Sc f Sc g 2δ . (1) | ρr s´ ρr s| ď 1 ?ρ 2 p ´ q Sc From now on we concentrate on estimating ρ g . Deﬁne the clumped square function Sq by r s

0 if x 0, ď Sq x x2 if 0 x 1, p q“ $ ď ď &’1 if x 1. ě It is not diﬃcult to check that Sq is 2-Lipschitz.%’ Corollary 5.12 together with Lemma 3.16 shows that for 1 Zp δ all ǫ 0, if τ, ǫ { then ą n ď 1 ρ 2 Ss Sc E E ? ´ ρ g ρ g Sq H?ρg Sq U?ρg Op ǫ O p ´ q . (2) | r s´ r s|“| νpnr p qs ´ Gpr p qs| “ p q` n ˆ ˙ Sc We would like to apply Borell’s theorem in order to bound ρ g , but g is not necessarily bounded by 0, 1 on Rn. In order to handle this, we deﬁne the functiong ˜ maxr 0s, min 1,g , which is bounded by 0, 1 . r s “ p p qq r s Let dist 0,1 be the function which measures the distance of a point x to the interval 0, 1 . The function r s r s 1 dist 0,1 is clearly 1-Lipschitz, and so Corollary 5.12 implies that under the stated assumptions on τ, n , we haver s E g g˜ E dist 0,1 g E dist 0,1 g E dist 0,1 g Op ǫ . Gpr| ´ |s “ Gpr r sp qs “ | νpnr r sp qs ´ Gpr r sp qs| “ p q

Since U?ρ is an averaging operator and Sq is 2-Lipschitz, we conclude that Sc Sc E E ρ g ρ g˜ Sq U?ρg Sq U?ρg˜ Op ǫ . (3) | r s´ r s|“| Gpr p qs ´ Gpr p qs| “ p q

Lemma 3.8 shows that EG g Eν g Eν f µ, and so EG g µ Op ǫ . Proposition 6.2 implies p r s“ pn r s“ pn r s“ | p r s´ |“ p q that Γρ E g˜ Γρ µ Op ǫ . Applying Borell’s theorem (Theorem 6.1), we deduce that p r sq ď p q` p q c S g˜ Γρ E g˜ Γρ µ Op ǫ . (4) ρr sď p r sq ď p q` p q Putting (1),(2),(3),(4) together, we conclude that

2 s 1 ?ρ ´ ?ρ S f Γρ µ Op ǫ O p ´ q 2δ . ρr sď p q` p q` n ` 1 ?ρ 2 ˆ ˙ p ´ q Taking δ ǫ, we obtain “ s 1 S f Γρ µ Op,ρ ǫ Oρ . ρr sď p q` p q` n ˆ ˙ 1 1 Zp ǫ The bounds on τ, now become τ, ǫ { , from which we can extract the theorem. n n ď 7 Bourgain’s theorem

Bourgain’s theorem in Gaussian space gives a lower bound on the tails of Boolean functions (in this section, Boolean means that the range of the function is 1 ). We quote its version from [21, Theorem 2.11]. t˘ u Theorem 7.1 (Bourgain). Let f : Rn 1 . For any k 1 we have, with respect to Gaussian measure N 0, 1 , Ñ t˘ u ě p q V k 2 f f ą Ω r s . } } “ ?k ˆ ˙

24 While the theorem is stated for N 0, 1 , it holds for Gp as well. Indeed, given a function f, deﬁne a new p q V V function g by g x f p 1 p x p . If x N 0, 1 then p 1 p x p Gp, and so f Gp g N 0,1 . p q“ p p ´ q ` q „ p q p ´ q ` „ r s “ r s p q f i G g i x f i p p x p g i i Our deﬁnition of “ fora p makes it clear that “ a“ 1 , where “ is the degree g f k 2 p qg “k 2 p p ´ q ` q f homogeneous part of . This implies that ą Gp ą N 0,1 . Therefore Bourgain’s theorem for and } } “} } p aq Gp follows from the theorem for g and N 0, 1 . Following closely the proof of [21, Theoremp q 3.1], we can prove a similar result for the slice.

n s k Theorem 7.2. Fix k 2. Let f : rpns 1 satisfy Infi f ď τ for all i n . For some constants 1 ě C Ñ t˘ Cu r s ď P r s Wp,k, C, if τ W ´ V f and n Wp,k V f then ď p,k r s ě` ˘ { r s V k 2 f f ą Ω r s . } } “ ?k ˆ ˙ s c The condition Inf f τ for all i n can be replaced by the condition Inf f µ τ for all i n . i r sď Pr s i r s p ď Pr s Proof. We treat f as a harmonic multilinear polynomial. Since f is Boolean, working over νpn we have

k 2 k 2 k k 2 k 2 k f ą f f ď f ď sgn f ď f ď 1 2 E f ď . } } “} ´ } ě} ´ p q} “} } ` ´ r| |s k 2 k 2 2 Lemma 5.2 shows that f ď f ď 1 Op k n . Since the absolute value function is 1-Lipschitz, } }Gp “} } p ˘ p { qq k K k K E k Theorem 5.8 applied to the parameter δ 0 shows that if τ Ip´ δ and n Ip δ then then νpn f ď k ą ď ě { | r| |s´ EG f ď Op δ . This shows that p r| |s| “ p q 2 k 2 k k 2 k f ą f ď sgn f ď Op δ . } }νpn ě} ´ p q}Gp ´ ` n ˆ ˙ k k 2 k 2 Let g sgn f ď . With respect to Gaussian measure Gp, f ď g gą Ω V g ?k , using Bourgain’s“ theoremp (Theoremq 7.1). Putting everything together,} we conclude´ } ě }that } “ p r s{ q

V 2 k 2 g Gp k f ą Ω r s Op δ . (5) } }νpn ě ?k ´ ` n ˆ ˙ ˆ ˙ V V It remains to lower bound g Gp . Note ﬁrst that over νpn, f 4Pr f 1 Pr f 1 , and so Pr f 1 , Pr f 1 V f r4.s We can furthermore assume thatr s “ r “ s r “ ´ s r “ s r “´ sě r s{ k 2 k 2 V Pr f ď 3 , Pr f ď 3 f 8, νpnr ě s νpnr ď´ sě νpnr s{

k 2 V V k 2 since if for example Pr f ď 3 f 8 then with probability at least f 8 we have f 1 and f ď 3 , k 2 r k ě2 sď1 Vr s{ V r s{ V “ ă and so f ą f f ď 9 f 8 Ω f . Corollary 5.9 applied with ǫ f 16 1 3 shows } } “} ´ } ě ¨ kr s{ “ p r sqk “ r s{ ď { that for an appropriate c, if τ X´ c and n cX then ď p { ě p k 1 k 1 V Pr f ď 3 , Pr f ď 3 f 16, Gp r ě s Gp r ď´ sě νpnr s{ V V V V and so Gp g 4 νpn f 16 1 νpn f 16 Ω νpn f . Combining this with (5) shows that under νpn, r sě r s{ p ´ r s{ q“ p r sq V 2 k 2 f k f ą Ω r s Op δ . (6) } } ě ?k ´ ` n ˆ ˙ ˆ ˙

Choosing δ cp V f ?k for an appropriate cp completes the proof. “ r s{ We do not attempt to match here [21, Theorem 3.2], which has the best constant in front of V f ?k. r s{

25 8 Kindler–Safra theorem

Theorem 7.2 implies a version of the Kindler–Safra theorem [22, 20], Theorem 8.5 below. We start by proving a structure theorem for almost degree k functions. We start with a hypercontractive estimate due to Lee and Yau [23] (see for example [13, Proposition 6.2]).

n Proposition 8.1. For every p there exists a constant rp such that for all functions f : r s R, Hr f 2 pn Ñ } p } ď f 4 3. } } { ` ˘ This implies the following dichotomy result.

n k 2 Lemma 8.2. Fix parameters p and k, and let f : rpns 1 satisfy f ą ǫ. For any i, j n , either s s Ñ t˘ u } } “ Pr s Inf f ǫ 2 or Inf f Jp,k, for some constant Jp,k. ij r sď { i,j r sě ` ˘ ij s 2 Proof. Let r rp be the parameter in Proposition 8.1. Let g f f p q 2, so that Infij f 4 g . Since “ 4 3 2 s “ p ´ q{ r s“ } } g x 0, 1 , g 4{3 g 4 Infij f . Proposition 8.1 therefore implies that p qPt ˘ u } } { “} } “ r s s 3 2 2 2 k 2 k k 2 4 Infij f { g 4 3 Hrg 2 Hrgď 2 r gď . p r sq “} } { ě} } ě} } ě } } ij k 2 k 2 Since g f f p q 2, we can bound gą f ą ǫ. Therefore “ p ´ q{ } } ď} } “ s 3 2 k 2 k s 4 Inf f { r g ǫ r 4 Inf f ǫ . p ij r sq ě p} } ´ q“ p ij r s´ q If 4 Infs f 2ǫ then 4 Infs f ǫ 4 Infs f 2 and so 4Infs f r2k 4. ij r są ij r s´ ą ij r s{ ij r sě { We need the following result, due to Wimmer [33, Proposition 5.3].

n Lemma 8.3 ([33, Proposition 5.3], [13, Lemma 5.2]). Let f : r s R. For every τ 0 there is a set pn Ñ ą J n of size O Inf s f τ such that Infs f τ whenever i, j J. Ďr s p r s{ q ij r să `R ˘ Combining Lemma 8.2 and Lemma 8.3, we deduce that bounded degree functions depend on a constant number of coordinates, the analog of [27, Theorem 1].

n Corollary 8.4. Fix parameters p and k. If f : r s 1 has degree k then f depends on Op,k 1 pn Ñ t˘ u p q coordinates (that is, f is invariant under permutations of all other coordinates). ` ˘

Proof. Apply Lemma 8.3 with τ Jp,k to obtain a set J of size O k Jp,k . Lemma 8.2 with ǫ 0 shows s “ p { q “ that for i, j J we have Infij f 0, and so f is invariant under permutations of coordinates outside of J. R r s “ Using Bourgain’s tail bound, we can deduce a stability version of Corollary 8.4, namely a Kindler–Safra theorem for the slice.

n k 2 Theorem 8.5. Fix the parameter k 2. Let f : rpns 1 satisfy f ą ǫ. There exists a function n ě Ñ t˘ u } } “ h: r s 1 of degree k depending on Ok,p 1 coordinates (that is, invariant under permutations of all pn Ñ t˘ u p q` ˘ other coordinates) such that ` ˘ 2 1 C 1 f h Op,k ǫ { , } ´ } “ ` n1 C ˆ { ˙ for some constant C.

k Proof. Let F f ď . We can assume that 2ǫ Jp,k 2, since otherwise the theorem is trivial. Apply “ ă { Lemma 8.3 to F with parameter τ Jp,k 2ǫ Jp,k 2, obtaining a set J of size O k τ Op,k 1 . It is not hard to check that “ ´ ą { p { q “ p q

s s s k 2 s Inf F Inf f Inf F 2 f ą Inf F 2ǫ. ij r sď ij r sď ij r s` } } “ ij r s`

26 s s s Therefore if i, j J then Infij f τ 2ǫ Jk,p, and so Lemma 8.2 shows that Infij F Infij f O ǫ . RJ r să ` “ r sď r s“ p q For x 0, 1 , let Gx and gx result from F and f (respectively) by restricting the coordinates in J to the Pt u J value x. It is not hard to check that PrS νpn S J x p Op J n | | Ωp,k 1 , as long as n Np,k „ r | “ s ě p ´ p| |{ qq “ p q s ě for some constant Np,k; if n Np,k then the theorem is trivial. We conclude that Infij Gx Op,k ǫ for 2 ď k 2 s r s “ p q all i, j J and Gx gx gxą Op,k ǫ . Together these imply that Infij gx Op,k ǫ for all i, j J, R s } ´ } “} } “ p q r s“ p q R and so Inf gx Op,k ǫ for all i J. i r s“ p q R We can assume that n J n 2 (otherwise the theorem is trivial) and that the skew px of the ´ | | ě { slice on which Gx,gx are deﬁned satisﬁes px p Op J n Θ p , and so Theorem 7.2 implies that s 1 V C “ ˘ V p| C|{ q “V p q k 2 either maxi Infi gx Wp,k´ gx , or n 2Wp,k gx , or gx O ?k gxą Op,k ǫ . Since s r s ą r s ă { r s r s “ p } } q “ p q maxi Inf gx Op,k ǫ , we conclude that i r s“ p q

1 C 1 V gx Op,k ǫ { . r s“ ` n1 C ˆ { ˙ E V Deﬁne a function g by g S gS J . The bound on gx implies p q“ r | s r s

2 1 C 1 f g Op,k ǫ { . } ´ } “ ` n1 C ˆ { ˙ If we let h sgn g then we obtain the desired bound f h 2 4 f g 2. “ } ´ } ď } ´ } It remains to show that h has degree k if ǫ is small enough and n is large enough. We can assume 2 k k 2 without loss of generality that J M , where M is the bound on J . We have f h f ą hą k 2 “r s | | } ´ } ě} ´ } ě hą ?ǫ . Therefore p} }´ q k 1 2C 1 hą ?ǫ Op,k ǫ { . } }ď ` ` n1 2C ˆ { ˙ On the other hand, we can write h as a Boolean function H of x1,...,xM . Lemma 4.1 shows that deg h k ď deg H, and so deg h k implies that deg H k. Corollary 4.4(5) implies that for large enough n, hą ą ą } }“ Ωp,H 1 . Since there are only ﬁnitely many Boolean functions on x1,...,xM which can play the role of H, we concludep q that if ǫ is small enough and n is large enough then deg h k. ď 1 C 1 C We conjecture that Theorem 8.5 holds with an error bound of Op,k ǫ rather than Op,k ǫ { 1 n { . p q p ` { q 9 t-Intersecting families

As an application of Theorem 8.5, we prove a stability result for the t-intersecting Erd˝os–Ko–Rado theorem, along the lines of Friedgut [17]. We start by stating the t-intersecting Erd˝os–Ko–Rado theorem, which was ﬁrst proved by Wilson [32].

n Theorem 9.1 ([32]). Let t 1, k t, and n t 1 k t 1 . Suppose that the family F rks is t-intersecting: every two setsě in F haveě at least ět points p ` qp in common.´ ` q Then: Ď ` ˘ n t (a) F k´t . | |ď ´ ` ˘ n t (b) If n t 1 k t 1 and F k´t then F is a t-star: a family of the form ą p ` qp ´ ` q | |“ ´ ` ˘ n F A r s : S A , S t. “t P k Ď u | |“ ˆ ˙ n t (c) If t 2, n t 1 k t 1 and F k´t then F is either a t-star or a t, 1 -Frankl family: ě “ p ` qp ´ ` q | |“ ´ p q n ` ˘ F A r s : A S t 1 , S t 2. “t P k | X |ě ` u | |“ ` ˆ ˙

27 The case t 1 is the original Erd˝os–Ko–Rado theorem [11]. Ahlswede and Khachatrian [1, 3] found the optimal t-intersecting“ families for all values of n,k,t. n t A stability version of Theorem 9.1 would state that if F k´t then F is close to a t-star. Frankl [14] proved an optimal such result for the case t 1. Friedgut| |« [17] proved´ a stability result for all t assuming ` ˘ that k n is bounded away from 1 t 1 . “ { {p ` q 1 n Theorem 9.2 ([17]). Let t 1, k t, λ, ζ 0, and λn k t 1 ζ n. Suppose F rks is a ě ě n t ąn ă ă p ` ´ q Ď t-intersecting family of measure F k´t ǫ k . Then there exists a family G which is a t-star such that | |“ ´ ´ ` ˘ ` ˘ F△G` ˘ | n | Ot,λ,ζ ǫ . k “ p q Careful inspection of Friedgut’s proof shows` ˘ that it is meaningful even for sub-constant ζ, but only as long as ζ ω 1 ?n . We prove a stability version of Theorem 9.1 which works all the way up to ζ 0. “ p { q “ Theorem 9.3. Let t 2, k t 1 and n t 1 k t 1 r, where r 0. Suppose that k n λ for ě ěn ` “ p ` qp ´ ` q` ąn t n { ě some λ 0. Suppose F rks is a t-intersecting family of measure F k´t ǫ k . Then there exists a familyąG which is a t-starĎ or a t, 1 -Frankl family such that | |“ ´ ´ ` ˘ p q ` ˘ ` ˘ 1 C F△G k { 1 C 1 | | Ot,λ max , 1 ǫ { , n “ r ` n1 C k ˜ ˜ˆ ˙ ¸ { ¸ for some constant C. ` ˘ C 1 Furthermore, there is a constant At,λ such that ǫ At,λ min r k, 1 ` implies that G is a t-star. ď p { q We do not know whether the error bound we obtain is optimal. We conjecture that Theorem 9.3 holds with an error bound of Ot,λ max k r, 1 ǫ . p p { q q Friedgut’s approach proceeds through the µp version of Theorem 9.1, first proved by Dinur and Safra [8] as a simple consequence of the work of Ahlswede and Khachatrian. The special case p 1 d (where d 3) also follows from earlier work of Ahlswede and Khachtrian [2], who found the optimal t“-agreeing{ familiesě in Zn d . Theorem 9.4 ([8],[17],[12]). Let t 1 and p 1 t 1 . Suppose that F 0, 1 n is t-intersecting. Then: ě ď {p ` q Ďt u t (a) µp F p [8]. p qď t (b) If p 1 t 1 and µp F p then F is a t-star [17]. ă {p ` q p q“ t (c) If t 2, p 1 t 1 and µp F p then F is either a t-star or a t, 1 -Frankl family [12]. ě “ {p ` q p q“ p q Friedgut [17] deduces his stability version of Theorem 9.1 from a stability version of Theorem 9.4. While Friedgut’s stability version of Theorem 9.4 is meaningful for all p 1 t 1 , his stability version of The- orem 9.1 is meaningful only for k n 1 t 1 ω 1 ?n . A moreă recent{p ` stabilityq result for compressed cross-t-intersecting families due to{ Frankl,ă {p Lee,` q´ Siggersp { andq Tokushige [15], using completely different tech- niques, also requires k n to be bounded away from 1 t 1 . Friedgut’s argument{ combines a spectral approach{p essentially` q due to Lovász [24] with the Kindler–Safra theorem [22, 20]. Using Theorem 8.5 instead of the Kindler–Safra theorem, we are able to obtain a stability result for the entire range of parameters of Theorem 9.1. We restrict ourselves to the case t 2. Our starting point is a calculation due to Wilson [32]. ě

n n Theorem 9.5 ([32]). Let t 2, k t 1, and n t 1 k t 1 . There exists an rks rks symmetric ě ě ` n ě p ` qp ´ ` q n ˆ matrix A such that ASS 1 for all S r s , AST 0 for all S T r s satisfying S T t, and for “ P k “ ‰ P k ` | ˘ X` |ě˘ ` ˘ ` ˘

28 n all functions f : r s R, k Ñ

k ` ˘ e Af λef “ , “ e 0 “ ÿ t 1 1 t 1 e ´ i k 1 i k e n k e i n k t i ´ λe 1 1 ´ ´ 1 ´ ´ ´ ´ ´ ` ´ ´ ` . “ ` p´ q p´ q k t i k e k t i 0 ˆ ˙ˆ ˙ˆ ˙ˆ ˙ ÿ“ ´ ´ ´ The eigenvalues λe satisfy the following properties:

1 n n t ´ (a) λ0 k k´t . “ ´ (b) λ1 ` ˘` λt˘ 0. “¨¨¨“ “ (c) λt 2 0, with equality if and only if n t 1 k t 1 . ` ě “ p ` qp ´ ` q (d) λt 1 λt 2 and λe λt 2 for e t 2. ` ą ` ą ` ą ` Wilson’s result actually needs n 2k, but this is implied by our stronger assumption k t 1 (Wilson only assumes that k t) since t 1ě k t 1 2k t 1 k t 1 0. ě ` ě p ` qp ´ ` q´ “ p ´ qp ´ p ` qq ě We need to know exact asymptotics of λt 2. `

Lemma 9.6. Let t 2, k t 1 and n t 1 ρ k t 1 , where ρ 0. Let λ λt 2 be the quantity deﬁned in Theoremě 9.5. Theně ` “ p ` ` qp ´ ` q ą “ ` λ Ωt min ρ, 1 , lim λ 1. ρ “ p p qq Ñ8 “ Proof. Wilson [32, (4.5)] gives the following alternative formula for λ:

t 1 k t t 1 ´ 2 t 1 i ´2 λ 1 ` ´ ` . “ ´ 2 i 2 i n k t i ˆ ˙ i 0 ˆ ˙ ´`i ´2˘` ÿ“ ` ` Algebraic manipulation shows that ` ˘

t 1 k t ´ t 1 i ´2 λ 1 i 1 ` ` . “ ´ p ` q i 2 n k t i i 0 ˆ ˙ ´`i ´2˘` ÿ“ ` ` Calculation shows that n k t t ρ k t 1 2t 1.` Therefore˘ ´ ´ “ p ` qp ´ ` q´ ` t 1 k t ´ t 1 i ´2 λ 1 i 1 ` ` . “ ´ p ` q i 2 t ρ k t 1 2t 1 i i 0 ˆ ˙ p ` qp ´`i` 2q´˘ ` ` ÿ“ ` ` ` ˘ This formula makes it clear that limρ λ 1, and that λ is an increasing function of ρ. Assume now that ρ 1. Then Ñ8 “ ď t 1 ´ t 1 i 2 1 λ 1 i 1 ` t ρ ´ ´ 1 Ot . “ ´ p ` q i 2 p ` q ˘ k i 0 ˆ ˙ ˆ ˆ ˙˙ ÿ“ `

29 Let us focus on the main term. Setting α 1 t ρ , we have “ {p ` q t 1 t 1 t 1 ´ t 1 i 2 ´ t 1 i 2 ´ t 1 i 2 i 1 ` α ` i 2 ` α ` ` α ` i i i i 0p ` q 2 “ i 0p ` q 2 ´ i 0 2 “ ˆ ` ˙ “ ˆ ` ˙ “ ˆ ` ˙ ÿ ÿ t 1 ÿt 1 ´ t i 2 ´ t 1 i 2 t 1 α ` ` α ` i i “ p ` q i 0 1 ´ i 0 2 “ ˆ ` ˙ “ ˆ ` ˙ ÿt t 1 ÿ t ` t 1 t 1 α αi ` αi i i “ p ` q i 1 ´ i 2 “ ˆ ˙ “ ˆ ˙ ÿ t ÿ t 1 t 1 α 1 α 1 1 α ` 1 t 1 α “ p ` q pp ` q ´ q ´ pp ` q ´ ´ p ` q q 1 1 α t 1 tα . “ ´ p ` q p ´ q Substituting α 1 t ρ , we obtain “ {p ` q t 1 t t ´ t 1 i 2 t ρ 1 ρ ρ t 1 ρ i 1 ` t ρ ´ ´ 1 p ` ` q 1 p ` ` q . p ` q i 2 p ` q “ ´ t ρ t t ρ “ ´ t ρ t 1 i 0 ˆ ˙ ` ÿ“ ` p ` q ` p ` q Therefore when ρ 1, ď ρ t 1 ρ t 1 λ p ` ` q Ot . “ t ρ t 1 ˘ k p ` q ` ˆ ˙ In particular, we can ﬁnd some constant Ct such that

t t 1 ρ Ct λ p ` ` q ρ . ě t ρ t 1 ´ k p ` q ` ˆ ˙

Therefore for 2Ct k ρ 1, we have λ Ωt ρ . Since λ is an increasing function of ρ, this shows that for { ď ď “ p q ρ 2Ct k, we have λ Ωt min ρ, 1 . ě { “ p p qq In order to finish the proof, we handle the case ρ Ct k. Consider n t 1 k t 1 1. The value of 1 λ in this case is ď { “ p ` qp ´ ` q` ´ t 1 k t ´ t 1 t 1 i ´2 1 λ ` ´ ` ´ “ i 2 i t k t 1 2t 2 i i 0 ˆ ˙ˆ ˙ p ´ `ìq´2˘ ` ` ÿ“ ` ` t 1 k t ´ t 1 t 1 ` i ´2 ˘ i 2 ` ´ ` 1 ` . “ i 2 i t k t 1 2t 1 i ´ t k t 1 2t 2 i i 0 ˆ ˙ˆ ˙ p ´ `ìq´2˘ ` ` ˆ ˙ ÿ“ ` ` p ´ ` q´ ` ` ` ˘ i 2 The value of the last expression without the correction term 1 ` is exactly 1 by Theorem 9.5, ´ t k t 1 2t 2 i and so p ´ ` q´ ` ` 2 1 λ Ωt . ě t k t 1 2t 2 “ k p ´ ` q´ ` ˆ ˙ Since λ is increasing in ρ, this shows that for all ρ 0 we have λ Ωt 1 k . If also ρ Ct k then this ą “ p { q ď { implies that λ Ωt ρ , finishing the proof. “ p q We need a similar result comparing the measures of t-stars and t, 1 -Frankl families. p q Lemma 9.7. Let t 2, k t 1 and n t 1 ρ k t t 1, where ρ 0. Let m be the measure of ě ě ` “ p ` ` qp ´ q` ` ą a t-star, and let m1 be the measure of a t, 1 -Frankl family. Then p q m m1 m m1 ´ Ωt min ρ, 1 , lim ´ 1. m ρ m “ p p qq Ñ8 “

30 Proof. We have n t n t 2 n t 2 m ´ , m1 t 2 ´ ´ ´ ´ . “ k t “ p ` q k t 1 ` k t 2 ˆ ´ ˙ ˆ ´ ´ ˙ ˆ ´ ´ ˙ Computation shows that

m m1 t 2 n k k t k t k t 1 ´ 1 p ` qp ´ qp ´ q ` p ´ qp ´ ´ q. m1 “ ´ n t n t 1 p ´ qp ´ ´ q If n t 1 k t 1 r then calculation shows that “ p ` qp ´ ` q`

m m1 r r t k t 1 n t 1 r r t k t ´ p ` p ´ q` q ´ ´ p ` p ´ 2qq. m1 “ n t n t 1 ě n t n t 1 p ´ qp ´ ´ q ´ p ´ ´ q Substituting r k t ρ, we obtain “ p ´ q 2 m m1 n t 1 k t ρ ρ t n t 1 ρ t ρ ´ ´ ´ p ´ 2q p ` q 2 ´ ´ p ` q 2 . m1 ě n t k t t 1 ρ “ n t t 1 ρ ´ p ´ q p ` ` q ´ p ` ` q

This shows that limρ m m1 m1 1. Since n t 1 k t 1 t 2 implies n t 1 n t 1 2, we also get Ñ8p ´ q{ “ ě p ` qp ´ ` qě ` p ´ ´ q{p ´ qě { m m1 ρ t ρ ´ p ` q 2 . m1 ě 2 t 1 ρ p ` ` q As ρ , the lower bound tends to 1 2, and in particular, we can find ct such that for ρ ct we have Ñ 8 { ě m m1 m1 1 3. When ρ ct, we clearly have m m1 m1 Ωt ρ , completing the proof. p ´ q{ ě { ď p ´ q{ “ p q t 2 The method of Lovász [24] as refined by Friedgut [17] allows us to deduce an upper bound on f ą for the characteristic function of a t-intersecting family. } }

n Lemma 9.8. Let t 2, k t 1 and n t 1 k t 1 r, where r 0. Let F rks be a t-intersecting family, and f its characteristicě ě ` function.“ Then p ` qp ´ ` q` ą Ď ` ˘ n t t 2 k k´t f ą O max , 1 m E f , where m ´ . } } “ r ¨ p ´ r sq “ n ˆ ˆ ˙˙ ` k ˘ 0 ` ˘ Proof. Let A be the matrix from Theorem 9.5. Since f “ E f , } }“ r s 2 t 2 E f f,Af λ0 E f λt 2 f ą . r s“x yě r s ` ` } } 1 1 This already implies that E f λ´ m. Since λ0 m´ and E f m, we conclude that r sď 0 “ “ r sď E 1 E 2 E 1 E E t 2 f m´ f f 1 m´ f m f f ą r s´ r s r sp ´ r sq ´ r s. } } ď λt 2 “ λt 2 ď λt 2 ` ` ` Lemma 9.6 completes the proof. In order to prove our stability result, we need a result on cross-intersecting families.

n n Theorem 9.9 ([16]). Let F ras and G rbs be cross-intersecting families: every set in F intersects every set in G. If n a b andĎ b a then Ď ě ` ` ě˘ ` ˘ n n a n F G ´ 1 . | | ` | |ď b ´ b ` ď b ˆ ˙ ˆ ˙ ˆ ˙ We can now prove our stability result.

31 Proof of Theorem 9.3. In what follows, all big O notations depend on t and λ. We can assume that n is large enough (as a function of t and λ), since otherwise the theorem is trivial. We use the parameter p k t 1 n which satisfies λ 2 p 1 t 1 . “ p ´ ` q{ { ă ă {p ` q E n t n Let f be the characteristic function of F, so that f m ǫ, where m k´t k . Lemma 9.8 shows t 2 r s“ ´ 2 “ ´ { that f ą O max k r, 1 ǫ, and so Theorem 8.5 shows that f g δ for the characteristic function g } } “ p p { qq } ´ } ď ` ˘ ` ˘ 1 C 1 C of some family G depending on J Jt coordinates, for some constant Jt, where δ O max k r { , 1 ǫ { 1 C “ “ p pp { q q ` 1 n { ; here we use the fact that λ k n 1 2. We want to show that if δ is small enough (as a function of{ t) thenq G must be a t-star or a t,ď1 -Frankl{ ď { family; if δ is large then the theorem becomes trivial. We start by showing that if δ pis smallq enough then G must be t-intersecting. Suppose without loss of J generality that G depends only on the first J coordinates. We will show that J G J 0, 1 must be t-intersecting. If J is not t-intersecting, then pick A, B J which are not t-intersecting,“ |r s withĎ t A u B . Let n J n J P 2 | | ě | | A S rk szrA s : A S F and B S rk szrB s : B S F . Since n t 1 k t 1 and k λn, “t P ´| | Y P u “t P ´| | Y P u ě p ` q ´ p ´ q ně J if n is large enough then k A k B n 2J, and so Theorem 9.9 shows that A B ´ . ` ˘ p ´ | |q ` p ´ | |q` ď ´˘ | | ` | |ď k B Therefore ´| | n J ` ˘ ´ 2 F△G k A A J A 1 f g | | ´| | p| | 1 p ´| | 1 O Ω 1 , } ´ } “ n ě n “ p ´ q ˘ p 1 p n “ p q k ` k ˘ ˆ ˆ p ´ q ˙˙ using Lemma 3.4 (for large` enough˘ n`) and˘ the fact that p λ 2. We conclude that if δ is small enough, ą { J G J must be t-intersecting. “Next,|r s we show that if δ is small enough then G must be either a t-star or a t, 1 -Frankl family. If G t p q is neither then µp J p for all 0 p 1 t 1 by Theorem 9.4, and in particular, since p λ 2, t p q ă ă ď {p ` q ą { µp J p γ for some γ 0; here we use the fact that there are finitely many t-intersecting families on J p qď ´ ą points. Since νk J µp J 1 O 1 n due to Lemma 3.4, for large enough n and small enough ǫ we have p q“ p qp ˘ p { qq f g 2 E f E g 2 γ 1 O 1 n ǫ 2 Ω 1 . } ´ } ě p r s´ r sq ě p p ˘ p { qq ´ q “ p q We deduce that if n is large enough and ǫ is small enough then G is either a t-star or a t, 1 -Frankl family. C 1 p q It remains to show that if ǫ At,λ min r k, 1 ` then G cannot be a t, 1 -Frankl family. Define ď p { q p q τ min r k, 1 . Let m1 be the measure of a t, 1 -Frankl family. Lemma 9.7 shows that m m1 Ω τ (since“ p p {λ 2q implies m Ω 1 ). Therefore ifp G qis a t, 1 -Frankl family then E g m Ω´ τ .“ On thep q ą { “ p q 1 C p1 Cq r s ď ´ p q other hand, E g E f δ m ǫ O ǫ τ { 1 n { . Put together, we obtain r sě r s´ “ ´ ´ pp { q ` { q 1 C 1 C Ω τ ǫ O ǫ τ { 1 n { . p qď ` pp { q ` { q Choose a constant c so that ǫ cτ implies ď 1 C 1 C Ω τ O ǫ τ { 1 n { ; p qď pp { q ` { q C C 1 if ǫ cτ then the theorem becomes trivial. The inequality implies that τ O ǫ τ and so τ ` O ǫ , ą “ p { q “ p q contradicting our assumption on ǫ for an appropriate choice of At,λ.

Our conjecture on the optimal error bound in Theorem 8.5 implies an error bound of Ot,λ max k r, 1 ǫ in Theorem 9.3. p p { q q

10 Open problems

Our work gives rise to several open questions.

1. Prove (or refute) an invariance principle comparing νpn and γp,p for arbitrary (non-harmonic) multilinear polynomials. 2. Prove a tight version of the Kindler–Safra theorem on the slice (Theorem 8.5).

32 3. The uniform distribution on the slice is an example of a negatively associated vector of random variables. Generalize the invariance principle to this setting. n 4. The slice rks can be thought of as a 2-coloring of n with a given histogram. Generalize the invariance principle to c-colorings with given histogram. r s ` ˘ n Fn 5. The slice rks has a q-analog: all k-dimensional subspaces of q for some prime power q. The analog Fn of the Boolean cube consists of all subspaces of q weighted according to their dimension. Generalize the invariance` ˘ principle to the q-analog, and determine the analog of Gaussian space.

Acknowledgements This paper started its life when all authors were members of a semester-long program on “Real Analysis in Computer Science” at the Simons Institute for Theory of Computing at U.C. Berkeley. The authors would like to thank the institute for enabling this work. Y.F. would like to mention that this material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, ﬁndings and conclusions or recommendations expressed in this material are those of the authors, and do not necessarily reﬂect the views of the National Science Foundation. The bulk of the work on this paper was done while at the Institute for Advanced Study, Princeton, NJ. E.M. would like to acknowledge the support of the following grants: NSF grants DMS 1106999 and CCF 1320105, DOD ONR grant N00014-14-1-0823, and grant 328025 from the Simons Foundation. K.W. would like to acknowledge the support of NSF grant CCF 1117079.

References

[1] Rudolf Ahlswede and Levon H. Khachatrian. The complete intersection theorem for systems of ﬁnite sets. Eur. J. Comb., 18(2):125–136, 1997. [2] Rudolf Ahlswede and Levon H. Khachatrian. The diametric theorem in Hamming spaces—optimal anticodes. Adv. Appl. Math., 20(4):429–449, 1998. [3] Rudolf Ahlswede and Levon H. Khachatrian. A pushing-pulling method: New proofs of intersection theorems. Combinatorica, 19(1):1–15, 1999. [4] Eiichi Bannai and Tatsuro Ito. Algebraic Combinatorics I: Association schemes. Benjamin/Cummings Pub. Co., 1984. [5] C. Borell. Geometric bounds on the Ornstein–Uhlenbeck velocity process. Z. Wahrsch. Verw. Gebiete, 70(1):1–13, 1985. [6] Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar. Direct sum testing. In ITCS 2015, 2015. [7] Anindya De, Elchanan Mossel, and Joe Neeman. Majority is stablest: discrete and SoS. In 45th ACM Symposium on Theory of Computing, pages 477–486, 2013. [8] Irit Dinur and Shmuel Safra. On the hardness of approximating minimum vertex cover. Ann. Math., 162(1):439–485, 2005. [9] Charles F. Dunkl. A Krawtchouk polynomial addition theorem and wreath products of symmetric groups. Indiana Univ. Math. J., 25:335–358, 1976. [10] Charles F. Dunkl. Orthogonal functions on some permutation groups. In Relations between combinatorics and other parts of mathematics, volume 34 of Proc. Symp. Pure Math., pages 129–147, Providence, RI, 1979. Amer. Math. Soc.

33 [11] Paul Erd˝os, Chao Ko, and Richard Rado. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford, 12(2):313–320, 1961. [12] Yuval Filmus. Spectral methods in extremal combinatorics. PhD thesis, University of Toronto, 2013. [13] Yuval Filmus. An orthogonal basis for functions over a slice of the boolean hypercube. Elec. J. Comb., 23(1):P1.23, 2016. [14] Péter Frankl. Erd˝os-Ko-Rado theorem with conditions on the maximal degree. J. Comb. Theory A, 46:252–263, 1987. [15] Peter Frankl, Sang June Lee, Mark Siggers, and Norihide Tokushige. An Erd˝os–Ko–Rado theorem for cross-t-intersecting families. J. Combin. Th., Ser. A, 128:207–249, 2014. [16] Peter Frankl and Norihide Tokushige. Some best-possible inequalities concerning cross-intersecting families. J. Combin. Th., Ser. A, 61:87–97, 1992. [17] Ehud Friedgut. On the measure of intersecting families, uniqueness and stability. Combinatorica, 28(5):503–528, 2008. [18] Marcus Isaksson and Elchanan Mossel. Maximally stable Gaussian partitions with discrete applications. Israel J. Math., 189(1):347–396, 2012. [19] Subhash Khot. Inapproximability of NP-complete problems, discrete fourier analysis, and geometry. In Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010. [20] Guy Kindler. Property testing, PCP and Juntas. PhD thesis, Tel-Aviv University, 2002. [21] Guy Kindler, Naomi Kirshner, and Ryan O’Donnell. Gaussian noise sensitivity and Fourier tails, 2014. Manuscript. [22] Guy Kindler and Shmuel Safra. Noise-resistant Boolean functions are juntas, 2004. Unpublished manuscript. [23] Tzong-Yau Lee and Horng-Tzer Yau. Logarithmic Sobolev inequality for some models of random walks. Ann. Prob., 26(4):1855–1873, 1998. [24] LászlóLovász. On the Shannon capacity of a graph. IEEE Trans. Inform. Theory, 25:1–7, 1979.

[25] Elchanan Mossel and Yuval Filmus. Harmonicity and invariance on slices of the Boolean cube. In 31st Conf. Comp. Comp., 2016. [26] Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low inﬂuences: Invariance and optimality. Ann. Math., 171:295–341, 2010. [27] Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. Comp. Comp., 4(4):301–313, 1994.

[28] Ryan O’Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. [29] Li Qiu and Xingzhi Zhan. On the span of Hadamard products of vectors. Linear Algebra Appl., (422):304–307, 2007. [30] Murali K. Srinivasan. Symmetric chains, Gelfand–Tsetlin chains, and the Terwilliger algebra of the binary Hamming scheme. J. Algebr. Comb., 34(2):301–322, 2011. [31] Hajime Tanaka. A note on the span of Hadamard products of vectors. Linear Algebra Appl., (430):865– 867, 2009.

34 [32] Richard M. Wilson. The exact bound in the Erd˝os-Ko-Rado theorem. Combinatorica, 4:247–257, 1984. [33] Karl Wimmer. Low inﬂuence functions over slices of the Boolean hypercube depend on few coordinates. In CCC, 2014.