Friedgut--Kalai--Naor Theorem for Slices of the Boolean Cube

Home , Boolean function

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n n r s x1,...,xn : xi k . k “ p q “ ˆ ˙ i“1 ÿ ( (Here n stands for the set 1,...,n .) Slices arise naturally in extremal combinatorics (the Erd˝os– r s t u Ko–Rado theorem), graph theory (G n, M graphs), and coding theory (constant-weight codes). p q They are also one of the simplest association schemes. Recently the study of analysis of Boolean functions on the slice has gained traction, as witnessed by several recent articles [22, 26, 8, 9, 10, 25]. Classical theorems in the area which have been generalized to the slice include, among else, the Kahn–Kalai–Linial theorem [15, 22], Friedgut’s junta theorem [11, 26, 8], and the Mossel–O’Donnell–Oleszkiewicz invariance principle [19, 9, 10]; see O’Donnell [21] for a description of these results (in their classical form). This paper continues the project of generalizing analysis of Boolean functions to functions on the slice by proving a slice analog of the fundamental structural result of Friedgut, Kalai and A. Naor [13], which states that if a Boolean function on the Boolean cube 0, 1 n is ǫ-close to an n t u affine function (a function of the form x1,...,xn c0 i“1 cixi) then it is O ǫ -close to one p q ÞÑ ` 2 p q of the functions 0, 1,xi, 1 xi. (Two functions f, g are ǫ-close if f g ǫ, where is the L2 ´ ř } ´ } ď }¨} norm.) Our main theorem states that for 2 k n 2 and p min k n, 1 k n , if a function ď ď ´ “ p { ´ { q f : rns 0, 1 is ǫ-close to an affine function for ǫ O p2 , then f is O ǫ -close to one of the k Ñ t u “ p q p q functions 0, 1,xi, 1 xi. For larger ǫ, we prove that either f or 1 f is O ǫ -close to maxi S xi ` ˘ ´ ´ p q P (when k n 2) or to mini S xi (when k n 2) for some set S of size S O ?ǫ p . ď { P ě { | |“ p { q (The logically inclined reader can read iPS xi for maxiPS xi and iPS xi for miniPS xi.) Ź Ž Comparison to other FKN theorems The original FKN theorem [13] states that if a function f : 0, 1 n 0, 1 is close to an affine function, then it is close to a dictatorship (a function t u Ñ t u depending on at most one coordinate). Kindler and Safra [18, 17] extended this theorem to the n µp-setting (the so-called biased Boolean cube), in which the distribution on 0, 1 is not uniform t u but is the product measure µp, in which Pr xi 1 p for i n . Their theorem states that for n r “ s “ P r s each p, if f : 0, 1 R is ǫ-close to an affine function (with respect to the µp measure), then f t u Ñ is O ǫ -close to a dictatorship; the hidden constant depends on p. (For more results in this vein, p q see [14, 20, 23, 24].) In contrast, our main theorem states that if p 1 2 and f : rns 0, 1 is ǫ-close to an affine ď { pn Ñ t u function then it is Cǫ-close to a dictatorship assuming ǫ O p2 ; for larger ǫ, we only guarantee “ p `2q ˘ that f depends on O ?ǫ p inputs. The restriction to ǫ O p is necessary, since the function 2 p { q “ p q max x1,x2 is p -close to the affine function x1 x2. However, for fixed p we can obtain a statement p q ` similar to that of the original FKN theorem by letting the constant C depend on p. As stated above, the error bound in the FKN theorem of Kindler and Safra depends on p. In contrast, the error bound in our theorem is uniform over all p, and this is why we cannot guarantee closeness to a dictatorship. Using our theorem as a black box, we prove a uniform version of the FKN theorem of Kindler and Safra, whose form is identical to that of our theorem for the slice. A similar situation occurs in the FKN theorem for permutations [6, 7]. We can think of the rns2 set Sn of permutations on n points as a certain subset of rns consisting of those sets whose projections to the individual coordinates are equal to n . The density parameter p thus has the r s ` ˘

2 2 value p n n 1 n, and as a result, given a function f : Sn 0, 1 which is close to an aﬃne “ { “ { Ñ t u function, we can only guarantee that it is close to a function of the form maxpi,jqPS xij. When f is balanced (E f 1 2), however, we are able to guarantee that f is close to a “permutation- r s « { dictatorship”; we refer the interested reader to [7].

On the proof Our proof uses a novel proof method, the random sub-cube method, which allows us to reduce the FKN theorem for the slice to the FKN theorem for the Boolean cube (see Keller [16] for a similar reduction from the biased µp measure on the Boolean cube to the uniform measure on the Boolean cube). The idea is to consider subsets of the slice which are isomorphic to a Boolean cube of dimension k (assuming k n 2): ď { a1, b1 ak, bk . t uˆ¨¨¨ˆt u We can apply the classical FKN theorem on each of these sub-cubes. Moreover, if we choose a1, b1, . . . , ak, bk at random, then a uniform point on a uniform sub-cube is just a uniform point on the slice, and this allows us to deduce our FKN theorem.

Applications Our main theorem has recently been used by Das and Tran [3] to determine the sharp threshold for the Erd˝os–Ko–Rado property on a random hypergraph, improving on an earlier result of Bollob´as et al. [2] which used the classical FKN theorem. For further work on the problem, see Devlin and Kahn [4].

Paper organization After some preliminary deﬁnitions appearing in Section 2, we formally state our main theorem in Section 3, where we also derive the uniform FKN theorem for the biased Boolean cube. The proof itself appears in Section 4.

Acknowledgements The author thanks Guy Kindler, Elchanan Mossel and Karl Wimmer for helpful discussions, Manh Tuan Tran for pointing out a mistake in an earlier version, and the anonymous reviewers for helpful suggestions.

2 Preliminaries

Notations We use the notations n 1,...,n and dist x,S miny S x y . r s “ t u p q“ P | ´ | A Boolean function is a 0, 1 -valued function. An affine function is a function of the form t u n ℓ x1,...,xn c0 cixi. p q“ ` i“1 ÿ Our main theorems involve the maximum and the minimum of a set of Boolean variables. We adopt the convention that max 0 and min 1. H“ H“ 2 2 For a function f on a finite domain D, the squared L2 norm of f is given by f E f , where } } “ r s the expectation is with respect to the uniform distribution over D. The squares norm satisfies a triangle inequality of the form f g 2 2 f 2 g 2 . } ` } ď p} } `} } q 2 We say that two functions f, g on the same domain are ǫ-close if f g ǫ. If f, g are } ´ } ď Boolean then they are ǫ-close iff Pr f g ǫ. If f is Boolean, g is real-value, and G is the r ‰ 2 s ď 2 Boolean function closest to g, then f G O f g . Note that G is obtained by “rounding” } ´ } “ p} ´ } q G to 0, 1 , that is G x argmin 0 1 x b . t u p q“ bPt , u | ´ |

3 Functions on the Boolean cube The Boolean cube is the set 0, 1 n. We identify functions on t u the Boolean cube with functions on the Boolean-valued variables x1,...,xn. Each function on the Boolean cube has a unique Fourier expansion

i S xi f x1,...,xn fˆ S χS, where χS 1 P . p q“ p q “ p´ q S n ř ÿĎr s Parseval’s identity states that f 2 fˆ S 2. } } “ S p q ř Functions on the slice For integers n 2 and 0 k n, the slice rns is deﬁned as ě ď ď k n ` ˘ n n r s x1,...,xn 0, 1 : xi k. k “ p q P t u “ ˆ ˙ i“1 ÿ ( Alternatively, we can think of rns as the collection of all subsets of n of size exactly k, using the k r s correspondence x1,...,xn i n : xi 1 . We use this correspondence freely in the paper. p q ÞÑ` t ˘P r s “ u Every aﬃne function on the slice has a unique representation of the form

n ℓ x1,...,xn cixi. p q“ i“1 ÿ The FKN theorem The Friedgut–Kalai–Naor theorem (FKN theorem for short) is the following result.

Theorem 2.1 (Friedgut–Kalai–Naor [13]). Suppose f : 0, 1 n 0, 1 is ǫ-close to an aﬃne t u Ñ t u function. Then f is O ǫ -close to one of the functions 0, 1,x1, 1 x1,...,xn, 1 xn . p q t ´ ´ u 3 Statement of main theorem

Our main theorem is a version of Theorem 2.1 for functions on a slice.

Theorem 3.1. Suppose f : rns 0, 1 is ǫ-close to an affine function, where 2 k n 2. k Ñ t u ď ď ´ Define p fi min k n, 1 k n . Then either f or 1 f is O ǫ -close to maxi S xi (when p 1 2) or p { ´ { q` ˘ ´ p q P ď { to mini S xi (when p 1 2) for some set S of size at most max 1, O ?ǫ p . P ě { p p { qq (We remind the reader that by convention, if S then maxiPS xi 0 and miniPS xi 1.) “H 2 “ “ The statement implies that for some constant C, if ǫ Cp then we are guaranteed that S 1, ă | |ď and so f can be approximated by a function of one of the forms 0, 1,xi, 1 xi. 2 ´ 2 The bound on the size of S is optimal: for ps o 1 , the function max x1,...,xs is O ps - “ p q p q pp q q close to the linear function x1 xs, but cannot be approximated using a smaller maximum `¨¨¨` 2 without incurring an error of Ω p ω ps . p q“ pp q q When k 0,n , the theorem is trivially true since every function is constant. When k P t u P 1,n 1 , the theorem is trivially true without the bound on S since for k 1 every function on trns ´ u rns | | “ is affine, and every Boolean function f on satisfies f max xi : f i 1 ; the case 1 1 “ t pt uq “ u k n 1 is similar. ` “˘ ´ ` ˘

4 3.1 Uniform FKN theorem for the Boolean cube Theorem 3.1 can be used to derive a uniform biased version of Theorem 2.1.

n Definition 3.2. For each n, the measure µp is a measure on 0, 1 whose value on the atom xi p1´xiq t u x1,...,xn is p i 1 p i . p q ř p ´ qř n Theorem 3.3. Suppose f : 0, 1 0, 1 is ǫ-close to an affine function with respect to the µp t u Ñ t u measure, for some p 0, 1 . Then with respect to the µp measure, either f or 1 f is O ǫ - P p q ´ p q close to maxi S xi (when p 1 2) or to mini S xi (when p 1 2) for some set S of size at most P ď { P ą { max 1, O ?ǫ min p, 1 p . p p { p ´ qqq Proof. Suppose that f is ǫ-close to the affine function ℓ. Let N be a large integer (we will take the rNs limit N later on), let kN tpNu, and consider the slice SN . It is not hard to check Ñ 8 “ “ kN that if A is chosen randomly from S, then the distribution of A n tends (as a function of N) to X r s` ˘ the distribution µp. Moreover, if we take pN min kN N, 1 kN N then pN min p, 1 p . “ p { ´ { q Ñ p ´ q Extend f to a function fN on the slice SN by taking fN x1,...,xN f x1,...,xn , and extend p q“ p2 q ℓ to a function ℓN in a similar way. The remarks above show that fN ℓN 2ǫ for large enough } ´ } ď N. Also, for large enough N, min kN ,N kN pN N 2 (since pN p). Therefore we can apply p ´ q“ ě Ñ Theorem 3.1 to deduce that for large enough N, either fN or 1 fN is O ǫ -close to a maximum ´ p q or a minimum of up to max 1, O ?ǫ pN max 1, O ?ǫ p inputs. p p { qq “ p p { qq Let gN denote the approximating function (the maximum or minimum of a small number of coordinates). If gN depends only on the first n coordinates, and g is the corresponding function on n 0, 1 , then f or 1 f is O ǫ -close to g, completing the proof. When gN depends on coordinates t u ´ p q beyond the first n, there exists a substitution σ to these coordinates such that fN σ or 1 fN σ | ´ | is O ǫ -close to gN σ. Since the number of substituted coordinates doesn’t depend on N, we can p q | complete the proof as before, noting that gN σ is either a maximum, a minimum, or a constant. | The proof of Theorem 3.3 is similar to an argument of Ahlswede and Khachatrian [1], in which n rNs the authors derive an Erd˝os–Ko–Rado theorem on 1,...,α from a similar theorem for ´1 . t u α N Another version of the same argument is due to Dinur and Safra [5], who derive an Erd˝os–Ko–Rado n rNs ` ˘ theorem on the Boolean cube 0, 1 with respect to µp from a similar theorem for . t u pN Arguments going in the other direction are also known (for example, Friedgut [12]), but are ` ˘ more complicated and sometimes result in degredation of parameters. This highlights the fact that Theorem 3.1 is more general than Theorem 3.3, in the sense that the former can be used to derive the latter, but not vice versa. On the other hand, while we derived Theorem 3.3 from Theorem 3.1, it is probably easier to prove it directly. In particular, the estimates on hypergeometric distributions which are necessary for the proof of Theorem 3.1 (for example, Lemma 4.7 below) would be replaced with similar but simpler estimates on geometric distributions.

3.2 The case ǫ “ 0

As a warm-up, we prove that the only Boolean aﬃne functions on the slice are 0, 1,xi, 1 xi. ´ Lemma 3.4. Suppose f : rns 0, 1 is aﬃne, where 2 k n 2. Then f 0, 1 or k Ñ t u ď ď ´ P t u f xi, 1 xi for some i. P t ´ u ` ˘

5 n Proof. Let f i“1 cixi. Without loss of generality, suppose that c1 min c1,...,cn . For any “rns “ p q i 1, let S be some set containing 1 but not i. Since f S△ 1, i f S ci c1 and f is ‰ P kř p t uq ´ p q“ ´ Boolean, we conclude that ci c1, c1 1 . If for all i 1 we have ci c1, then f 0, 1 . So we ` ˘ P t ` u ‰ “ P t u can assume that I0 i n : ci c1 and I1 i n : ci c1 1 are both non-empty. “ t P r s “ u “ t P r s “ ` u We claim that either I0 1 or I1 1. Otherwise, suppose without loss of generality that | | “ | | “ rns 1, 2 I0 and 3, 4 I1 (here we are using the fact that 2 k n 2). Let S be some P P ď ď ´ P k set containing both 1, 2 but neither 3, 4. Then f S△ 1, 2, 3, 4 f S c3 c4 c1 c2 2, p t uq ´ p q “ ` ´` ˘´ “ contradicting the fact that f is Boolean. This shows that either I0 1 or I1 1. If I0 1 | | “ | | “ “ t u then n n f c1x1 c1 1 xi c1 1 xi x1 c1 1 k x1, “ ` i“2p ` q “ p ` q i“1 ´ “ p ` q ´ ÿ ÿ and since f is Boolean, necessarily f 1 x1. Similarly, if I1 i then we get f xi. “ ´ “ t u “ 4 Proof of main theorem

4.1 Proof overview rns rns Since every function on k is equivalent to a function on n´k , and the equivalence preserves affine functions, it suffices to consider the case k n 2. ` ˘ ď { ` ˘ For the rest of this section, we make the assumption that 2 k n 2 and fix the following ď ď { notation: • p k n 1 2. “ { ď { • f : rns 0, 1 is a Boolean function. k Ñ t u n 2 • ℓ ` ˘ 1 cixi is an affine function satisfying f ℓ ǫ. “ i“ } ´ } ď We first explain the proof of Theorem 3.1 in the easier case ǫ p 128. Extending Lemma 3.4, ř ă { we show that the coefficients c1,...,cn are close to two values α, α 1, say most of them close to α. ` We define di ci or di ci 1 in such a way that d1,...,dn are all close to α, and let r dixi. “ “ ´ “ i Note that h ℓ r is of the form iPS xi. Applying the classical Friedgut–Kalai–Naor theorem (in “ ´ rns rns k ř the form of Lemma 4.2) to a random sub-cube of (a subset of of the form 1 ai, bi , ř k k i“ t u where a1, b1, . . . , ak, bk are distinct elements of n ), we deduce that r s ` ˘ ` ˘ Ś 2 2 k E di dj k E dist ci cj, 0, 1 O ǫ . i‰jp ´ q “ i‰j p ´ t ˘ uq “ p q 2 A simple calculation shows that V r k Ei j di dj O ǫ , and so, putting m E r, we get ď ‰ p ´ q “ p q “ that 2 2 2 f h m 2 f ℓ 2 r m O ǫ . } ´ p ` q} ď } ´ } ` } ´ } “ p q This means that f is close to the function H obtained from rounding h m to 0, 1 . The proof is ` t u complete by showing that the only way a function of the form h m is close to a Boolean function ` is when H maxiPS xi (this includes the case H 0) or H 1. “ “ “ 2 When ǫ p 128 we cannot deduce that di dj dist ci cj, 0, 1 . Instead, we start by ě { ´ “ p ´ t ˘ uq showing that all but an O ǫ fraction of the coefficients c1,...,cn are concentrated around three p q values α 1, α, α 1. This allows us to approximate f by a function of the form xi ´ ` iPS` ´ xi m. Further arguments show that one of the first two summands can be bounded by iPS´ ` ř O ǫ in expectation, and the proof is completed as before. řp q

6 4.2 First steps 4.2.1 Concentration of coeﬃcients

We start by showing that for small ǫ p, the coefficients c1,...,cn are all concentrated around two { values α, α 1. ` Lemma 4.1. There exist c, d satisfying c d 1 and a subset S n of size S n 2 such that 2 | ´ |“ 2 Ď r s | |ď { for all j S, cj c 8ǫ p, and for all j S, cj d 8ǫ p. P | ´ | ď { R | ´ | ď { Proof. Suppose, without loss of generality, that c1 min c1,...,cn , and fix i 1. For any “ p q ‰ T rns , define P k 2 2 ` ˘ ∆ dist ℓ T , 0, 1 dist ℓ T △ 1, i , 0, 1 “ p p q t uq2 ` p p t uq t uq 2 dist ℓ T , 0, 1 dist ℓ T ci c1, 0, 1 . “ p p q t uq ` p p q` ´ t uq

Let r ℓ T and δ ci c1 0. Suppose that ℓ T is closer to a 0, 1 and ℓ T △ 1, i is closer “ p q “ ´ ě p q P t u p t uq to b 0, 1 , where a b. Then P t u ď 2 2 2 δ b a ∆ r a r δ b p ´ ` q , “ p ´ q ` p ` ´ q ě 2 using the inequality α2 β2 α β 2 2 with α a r and β r δ b. Since b a 0, 1 , ` ě p ` q { “ ´ “ ` ´ ´ P t u we deduce that

2 2 1 2 dist ℓ T , 0, 1 dist ℓ T △ 1, i , 0, 1 dist ci c1, 0, 1 . p p q t uq ` p p t uq t uq ě 2 p ´ t uq Taking expectation over random T , we conclude that

1 2 dist ci c1, 0, 1 Pr 1 T, i T 2ǫ. 2 p ´ t uq r P R sď

rns kpn´kq Since a random T k contains 1 but not i with probability npn´1q p 1 p p 2, we conclude P 2 ě p ´ qě { that dist ci c1, 0, 1 8ǫ p. p ´ t `uq˘ ď { For α 0, 1 , let Iα be the set of indices i such that ci c1 is closer to α. The lemma now P t u ´ follows by either taking S I0, c c1 and d c1 1 or S I1, c c1 1 and d c1. “ “ “ ` “ “ ` “ 4.2.2 The random sub-cube argument

Unconditionally, the values ci are on average either close to one another or at distance roughly 1. We show this by taking a random sub-cube of dimension k, and applying the classical Friedgut– Kalai–Naor theorem to the restrictions of f and ℓ to the sub-cube. First, we need the following consequence of the FKN theorem.

Lemma 4.2. Suppose f : 0, 1 n 0, 1 is ǫ-close to an aﬃne function ℓ: 0, 1 n R. Then t u Ñ t u t u Ñ n 2 dist 2ℓˆ i , 0, 1 O ǫ . i“1 p pt uq t ˘ uq “ p q ÿ

7 Proof. Theorem 2.1 shows that f is O ǫ -close to some function g of one of the forms 0, 1,xi, 1 xi. p q ´ The Fourier expansions of these functions are, respectively, 1 1 1 1 0, 1, 1 xi , 1 xi . 2 ´ 2p´ q 2 ` 2p´ q In particular,g ˆ i 0, 1 2 for all i n . pt uq P t ˘ { u P r s2 2 2 The triangle inequality shows that g ℓ 2 g f 2 f ℓ O ǫ . On the other hand, } ´ } ď } ´ } ` } ´ } “ p q Parseval’s identity shows that

n n 2 2 2 2 g ℓ 2ˆg i 2ℓˆ i dist 2ℓˆ i , 0, 1 . } p ´ q} ě i“1r pt uq ´ pt uqs ě i“1 p pt uq t ˘ uq ÿ ÿ We can now apply the random sub-cube argument.

Lemma 4.3. We have 2 k E dist ci cj, 0, 1 O ǫ . i‰j p ´ t ˘ uq “ p q

Proof. Let a1, b1, . . . , ak, bk be 2k distinct random indices taken from n , and deﬁne r s n D a1, b1 ak, bk r s . “ t uˆ¨¨¨ˆt uĎ k ˆ ˙ Clearly 2 2 E f D ℓ D f ℓ ǫ. D } | ´ | } “} ´ } ď k Using the mapping a1, b1 ak, bk 0, 1 0, 1 0, 1 , we can think of D as a t uˆ¨¨¨ˆt u « t uˆ¨¨¨ˆt u “ t u k k-dimensional Boolean cube. Under this encoding, f D is a Boolean function 0, 1 0, 1 , and | t u Ñ t u n n

ℓ D y1,...,yk cai cbi cai yi. | p q“ ` p ´ q i“1 i“1 ÿ ÿ

yi Since yi 1 1 2, we see that 2ℓ D i cai cbi . Lemma 4.2 therefore shows that “ p ´ p´ q q{ | pt uq “ ´ k x 2 2 dist cbi cai , 0, 1 O f D ℓ D . p ´ t ˘ uq “ p} | ´ | } q i“1 ÿ The lemma now follows by taking the expectation over the choice of D.

4.2.3 A variance formula An estimate of the type given by Lemma 4.3 is useful since it can potentially bound the variance of ℓ, as the following lemma shows.

Lemma 4.4. For r dixi we have “ i ř k n k 2 2 V r p ´ q E di dj k E di dj . “ 2 n 2 i‰jp ´ q ď i‰jp ´ q p ´ q

8 Proof. By shifting all coeﬃcients di, we can assume without loss of generality that E r 0 and so n E “ E 2 i“1 di 0 (this does not aﬀect the quantities di dj). For every i j we have xi xi p “ k´1 ´ ‰ “ “ and E xixj p. Therefore ř “ n´1

2 2 k 1 k 1 2 k n k 2 V r E r p d ´ p d d p 1 ´ d p ´ q d . i n 1 i j n 1 i n n 1 i “ “ i ` i j‰i “ ´ i “ i ÿ ´ ÿ ÿ ˆ ´ ˙ ÿ p ´ q ÿ On the other hand,

E 2 2 2 2 1 2 2 2 n 2 2 di dj di di dj 1 di p ´ q di . i‰j n n n 1 n 1 n n n 1 p ´ q “ i ´ i j‰i “ ´ i “ i ÿ p ´ q ÿ ÿ ˆ ´ ˙ ÿ p ´ q ÿ We conclude that V k n k 2 k n k E 2 r p ´ q di p ´ q di dj . n n 1 2 n 2 i‰j “ i “ p ´ q p ´ q ÿ p ´ q As a corollary, we obtain the following criterion for approximating f by an aﬃne function.

2 Corollary 4.5. Suppose d1,...,dn are coefficients satisfying k Ei‰j di dj O ǫ , and define 2 p ´ q “ p q g ci di xi. Then for some m, f g m O ǫ . “ ip ´ q } ´ p ` q} “ p q Proof.ř Define r ℓ g dixi. Lemma 4.4 shows that V r O ǫ , and so “ ´ “ i “ p q 2 2 2 fř g E r 2 f ℓ 2 r E r O ǫ . } ´ p ` q} ď } ´ } ` } ´ } “ p q 4.3 The case ǫ ă p{128 As an application of the corollary, we show that when ǫ p 128, the function f can be approximated ă { by a function of the form xi m. The condition ǫ p 128 ensures that the estimate of ˘ iPS ` ă { Lemma 4.1 is strong enough to deduce di dj dist ci cj, 0, 1 for appropriate di chosen ř | ´ | “ p ´ t ˘ uq according to the lemma.

Lemma 4.6. If ǫ p 128 then there exist δ 1 , real m, and a subset S n of size at most ă { 2 P t˘ u Ď r s n 2, such that f δ xi m O ǫ . { } ´ p iPS ` q} “ p q Proof. Lemma 4.1 showsř that for some c, d satisfying c d 1 there exists a subset S n of size 2 | ´ |“ 2 Ď r s at most n 2 such that for i S, ci c 8ǫ p, and for i S, ci d 8ǫ p. Let δ d c 1 , { P | ´ | ď { R | ´ | ă { “ ´ P t˘ u and deﬁne r ℓ δ xi. Note that r dixi, where di ci δ for i S and di ci for i S. “ ` iPS “ i “ ` P “ R In both cases, di d 8ǫ p 1 4, and so di dj 1 2 for all i, j. Since di dj ci cj κ | ´ ř|ď { ă { ř | ´ |ă { ´ “ ´ ` for some κ 0, 1 , this shows that dist ci cj, 0, 1 di dj , and so Lemma 4.3 implies P t ˘ u a p ´ t ˘ uq “ | ´ | that 2 k E di dj O ǫ , i‰jp ´ q “ p q The lemma now follows from Corollary 4.5.

The next step is to determine when a function of the form xi m can be close to ˘ iPS ` Boolean. The idea is to analyze the hypergeometric random variable xi using the following ř iPS lemma, whose somewhat technical proof appears in Section 4.5. ř

9 Lemma 4.7. There exists a constant γ0 0 such that the following holds, for all k n 2. Consider ą ď { the random variable X xi. If t n 2 and Pr X m,m 1 1 γ0 for some m then “ iPrts ď { r P t ` us ě ´ Pr X 0 Ω 1 and t 3 2 p´1. r “ s“ p q ďř p { q The lemma (and a few of its consequences) doesn’t use the condition k 2, and we will need ě this fact in the following section. We can now determine which functions of the form xi m are close to Boolean. It is ˘ iPS ` enough to consider the case xi m, the other case following by considering the function 1 f iPS ` ř ´ instead. ř Lemma 4.8. There exists a constant γ1 0 such that the following holds, for all k n 2. If 2 ą ď2 { γ ﬁ f iPS xi m γ1, where S n 2, then for some µ 0, 1 , f iPS xi µ O γ . } ´p ` q} ď 1 | |ď { P t u } ´p ` q} “ p q Furthermore, S 3 2 p´ and m µ O ?γ . ř | | ď p { q | ´ |“ p q ř

Proof. Let X iPS xi, and define µ to be the integer closest to m. Since Pr X µ, 1 µ “ 2 r R t´ ´ us ď 4 f iPS xi m 4γ1, Lemma 4.7 shows (assuming 4γ1 γ0) that Pr X 0 Ω 1 } ´ p `ř 1q} ď ď r “ s “ p q and S 3 2 p´ . This implies that µ 0, 1 , since otherwise γ Ω Pr X 0 Ω 1 . | |ř ď p { q 2 P t u “ p r “ sq “ p q Furthermore, γ Ω m µ and so m µ O ?γ . For any non-zero integer z, we have “ p| ´ | q | ´ | “ p q z µ z m m µ 2 z m , since µ is the integer closest to m. This shows that | ´ | ď | ´ | `2 | ´ | ď | ´ | f xi µ 4γ. } ´ p iPS ` q} ď Corollaryř 4.9. Let γ1 be the constant in Lemma 4.8. For all k n 2, the following holds. If 2 ď { 2 γ fi f iPS xi m γ1, where S n 2, then for some µ k, 1 k , f iPS xi µ } ´p ` q} ď | 1|ě { P t´ ´ u } ´p ` q} “ O γ . Furthermore, S n 3 2 p´ and m µ O ?γ . p q ř | |ě ´ p { q | ´ |“ p q ř Proof. Suppose that S n 2. Since xi k xi we have | |ą { iPS “ ´ iRS ř ř 1 f xi 1 k m f xi m . p ´ q ´ p ` ´ ´ q“´r ´ p ` qs iRS iPS ÿ ÿ Therefore Lemma 4.8, applied to f 1 fi 1 f (a Boolean function), S1 fi S, and m1 fi 1 k m, ´ ´ ´ shows that for some µ1 0, 1 , P t u 1 2 1 f xi µ O γ . }p ´ q ´ p ` q} “ p q iRS ÿ 1 2 ´1 Taking µ 1 k µ , we deduce that f xi µ O γ . Furthermore, n S 3 2 p “ ´ ´ } ´p iPS ` q} “ p q ´| | ď p { q and m µ m1 µ1 O ?γ . | ´ | “ | ´ |“ p q ř Putting Lemma 4.6 and Lemma 4.8 together, we get that f or 1 f is O ǫ -approximated by ´1 ´ p q a function of the form maxi S xi, where S O p . (When µ 1, f is close to a constant.) In P | |“ p q “ order to improve the bound on S , we estimate the probability that xi 2. | | iPS ě Lemma 4.10. Let S be a subset of n of size t fi S 3 2 p´1. Ifřt 2 then r s | | ď p { q ě

2 Pr xi 2 Ω pt . ě “ pp q q «iPS ﬀ ÿ

10 Proof. Let p1 k´1 p 2 (using k 2) and p2 k´2 p. The inclusion-exclusion principle “ n´1 ě { ě “ n´2 ď shows that

t t Pr xi 2 Pr x1 x2 1 Pr x1 x2 x3 1 ě ě 2 r “ “ s´ 3 r “ “ “ s «iPS ﬀ ÿ ˆ ˙ ˆ ˙ t tp2 pp1 1 “ 2 ´ 3 ˆ ˙ ˆ ˙ t2 p2 1 tp 2 p q . ě 2 2 2 “ 8 We can now prove Theorem 3.1 when ǫ p 128. ă { Lemma 4.11. Suppose f : rns 0, 1 is ǫ-close to an aﬃne function, and let p min k n, 1 k Ñ t u “ p { ´ k n . If 2 k n 2 and ǫ p 128 then either f or 1 f is O ǫ -close to maxi S xi for some { q ď ď ´ ` ă˘ { ´ p q P set S of size at most max 1, ?ǫ p . p { q

Proof. Lemma 4.6 shows that for some real m and set S of size at most n 2 we have f δ iPS xi 2 { } ´p ` m O ǫ . For simplicity, assume that δ 1 (when δ 1, consider 1 f instead of f). q} “ p q “ 2 “ ´ ´ ř Lemma 4.8 then implies that f xi µ O ǫ for some µ 0, 1 , assuming ǫ is small } ´ p iPS ` q} “ p q P t u enough (otherwise the lemma is trivially true, since every Boolean function is 1-close to the constant ř 1 0 function), and moreover S 3 2 p´ . | | ď p { q 2 Suppose ﬁrst that µ 0. In this case, if S 2 then Lemma 4.10 implies that p S O ǫ “ | |ě p | |q “ p q and so S O ?ǫ p . The function g maxiPS xi results from rounding h ﬁ iPS xi to Boolean, | |“ 2p { q 2 “ 2 and so f g O f h O ǫ . When µ 1, we similarly get f 1 O ǫ . } ´ } “ p} ´ } q“ p q “ } ´ } ř“ p q 4.4 The case ǫ “ Ωppq We move on to the case ǫ Ω p . In this case the analog of Lemma 4.6 states that f can be “ p q approximated by a function of the form xi xi m, where at least one of S ,S is iPS` ´ iPS´ ` ` ´ small. ř ř |S`| |S´| Lemma 4.12. There exist real m and two subsets S`,S´ n satisfying n n O ǫ k such 2 Ď r s “ p { q that f xi xi m O ǫ . } ´ p iPS` ´ iPS´ ` q} “ p q 2 Proof. Lemmař 4.3 showsř that k Ej i dist ci cj, 0, 1 O ǫ . This implies that for some ‰ p ´ t ˘ uq “ p q i0 n , P r s 2 k E dist ci0 cj, 0, 1 O ǫ . j‰i0 p ´ t ˘ uq “ p q

We partition the coordinates in n into four sets. For δ 0, 1 , we let Sδ j n : cj ci0 r s P t ˘ u E “ t P r s | ´ 2 ´ δ 1 4 , and we put the rest of the coordinates in a set R. Since j‰i0 dist ci0 cj, 0, 1 | ă|R| { u |R| 2 |S´1| |S`1|p ´ t ˘ uq “ Ω , we conclude that O ǫ k . Since Ei j ci cj Ω , we conclude that p n q n “ p { q ‰ p ´ q “ p n n q |S´1| |S`1| O ǫ k . n n “ p { q Deﬁne now di ci 1 for i S 1, di ci 1 for i S 1, and di ci otherwise. When “ ´ P ` “ ` P ´ “ i, j S0 S 1 or i, j S0 S 1, we get di dj 1 2, and since di dj ci cj κ P Y ` P Y ´ | ´ | ă { ´ “ ´ ` for some κ 0, 1 , we conclude that dist ci cj, 0, 1 di dj . For all i, j we claim that 2 P t ˘ u 2 p ´ t ˘ uq “ | ´ | di dj 7 dist ci cj, 0, 1 16. Indeed, for some κ 0, 1, 2 we have di dj ci cj κ. p ´ q ď p ´ t ˘ uq ` P t ˘ ˘ u ´ “ ´ `

11 2 2 If ci cj 2 then di dj 16, whereas if ci cj 2, say ci cj 2, then ci cj 1 ci cj 1 | ´ |ď p ´ q ď | ´ |ě ´ ě ´ ´ ď p ´ ´ q and so

2 2 di dj ci cj 1 κ 1 p ´ q “ pp ´ ´ q ` p `2 qq 2 dist ci cj, 0, 1 2 ci cj 1 κ 1 κ 1 “ p ´ t ˘ uq 2` p ´ ´ qp ` q ` p ` q 7 dist ci cj, 0, 1 9. ď p ´ t ˘ uq `

Call a pair of indices i, j good if i, j S0 S`1 or i, j S0 S´1, and note that the probability P Y 1 1 P Y that i, j is bad (not good) is at most 2 |R| 2 |S` | |S´ | O ǫ k . This shows that n ` n n “ p { q 2 2 k E di dj 7k E dist ci cj, 0, 1 16k Pr i, j bad O ǫ . i‰jp ´ q ď i‰j p ´ t ˘ uq ` r s“ p q Corollary 4.5 now completes the proof.

An argument similar to the one in Lemma 4.11 completes the proof of the theorem.

Proof of Theorem 3.1. If ǫ p 128 then the result follows from Lemma 4.11, so we can assume ď { that ǫ p 128, and in particular we can assume that p 1 5, since otherwise ǫ 1 640 and so the ą { ď { ą { result is trivial (since every Boolean function is 1-close to the constant 0 function). In several other places in the proof we also assume that ǫ is small enough (smaller than some universal constant independent of p); otherwise the result is trivial. Lemma 4.12 shows that for some real m and sets S ,S it holds that f xi xi ` ´ } ´p iPS` ´ iPS´ ` m 2 O ǫ . By possibly replacing f by 1 f, we can assume that S S . In particular, q} “ p q ´ | ´| ďř | `| ř S n 2, and so k n S 2p 1 2. Note that S could be empty. | ´|ď { {p ´ | ´|q ď ă { ´ We now consider two diﬀerent cases: S n S 2 and S n S 2. | `| ď p ´ | ´|q{ | `| ě p ´ | ´|q{

Case 1: S` n S´ 2. Consider some setting of the variables in S´ which sets w of them 1 | | ď p ´ | |q{ rn s 1 to 1. This setting reduces the original slice to a slice Z isomorphic to k1 , where n n S´ 1 1 k1 1 k “ ´ | | and k k w. The corresponding p 1 satisﬁes p 1 2. Lemma 4.8 applied with n n´|S´| ` ˘ “ ´ “ ď ă { 2 m ﬁ m iPS´ xi shows that for each such setting, either f iPS` xi iPS´ xi m Z Ω 1 ´ } ´ p ´1 ` q} “ p q or dist m xi, 0, 1 1 4 (note that the lemma works even if k 2). In the latter case, p ř´ iPS´ t uq ď { ř ăř we say that w is good. ř If no w is good then ǫ Ω 1 , so by assuming that ǫ is small enough we can guarantee that “ p q some w is good. On the other hand, the condition on m shows that at most two values w0, w0 1 ` are good. This implies that Pr xi w0, w0 1 1 O ǫ , and so for small enough ǫ, r iPS´ P t ` us ě ´ p q Lemma 4.7 shows that Pr xi 0 Ω 1 . Therefore we can assume that w0 0. r iPS´ř “ s“ p q “ Let w“W denote the norm restricted to inputs in which iPS´ xi W (we similarly use }¨} ř “ 2 Prw w ). Since Pr xi 0 Ω 1 , we must have f xi m 0 O ǫ . “ r¨s r iPS´ “ s “ p q } ř´ p iPS` ` q}w“ “ p q Lemma 4.8 shows that for some µ 0, 1 , also f x µ 2 O ǫ , and moreover ř iPS` i řw“0 1´P1 t u 1 } ´ p1 ` q} “ p q m µ 1 4 and S` 3 2 p , where p k n p. Lemma 4.10 implies that in fact | ´ |2 ď { | | ď p { q “ { řě p1 S O ǫ , and so S O ?ǫ p1 O ?ǫ p . p | `|q “ p q | `|“ p { q“ p { q We now consider two subcases: µ 0 and µ 1. “ “

12 Case 1(a): µ 0. Since g ﬁ maxiPS` xi is the result of rounding iPS` xi µ to Boolean, we “ 2 ` conclude that f g 0 O ǫ . Since m 1 4, it cannot be the case that dist m 1, 0, 1 } ´ }w“ “ p q | |ď { ř p ´ t uq ď 1 4, and so 1 is not good. In other words, Pr xi 0 1 O ǫ . This implies that { r iPS´ “ s ě ´ p q f g 2 O ǫ , completing the proof in this case. } ´ } “ p q ř

Case 1(b): µ 1. In this case Prw 0 xi 0 1 O ǫ , since xi µ f x “ “ r iPS` “ sě ´ p q | iPS` ` ´ p q| ě xi. This implies that Pr xi 0 1 O ǫ , since the assumption xi 0 only iPS` r iPS` “ řs ě ´ p q ř iPS´ “ makes it harder for xi to vanish. Since S` S´ , this implies that Pr xi 0 ř iPS` ř | | ě | | řr iPS´ “ s ě 1 O ǫ , for similar reasons. We conclude that h ﬁ xi xi vanishes with probability ´ p q ř iPS` ´ iPS´ ř 1 O ǫ . Since f h m 2 O ǫ , it follows that Pr f ν 1 O ǫ , where ν is the rounding ´ p q } ´p ` q}2 “ p q ř r “ s“ř ´ p q of m to 0, 1 . Thus f ν O ǫ , completing the proof in this case. t u } ´ } “ p q

Case 2: S` n S´ 2. In the setup of the first case, applying Corollary 4.9 instead | | ě p ´ | |q{ 2 of Lemma 4.8 allows us to conclude that either f iPS` xi iPS´ xi m Z Ω 1 or 1 1 } ´ p ´ ` q} “ p q dist m xi k , 0, 1 1 4. Since k k xi, we can rewrite the latter condition p ´ iPS´ ` t uq ď { “ ´ iPSř´ ř as dist m 2 xi k, 0, 1 1 4. As before, in the latter case we say that w xi is p ř´ iPS´ ` t uq ď { ř “ iPS´ good. ř ř In contrast to the situation in Case 1, here at most one (and so exactly one) w can be good. Lemma 4.7 implies that w 0, and so Pr xi 0 1 O ǫ . This implies that f “ r iPS´ “ s ě ´ p q } ´ x m 2 O ǫ . Corollary 4.9 implies that for some µ 0, 1 we have f x iP|S`| i w“0 ř iPS` i p 2 ` q} “ p q 1 1´1 P t u } ´p ´ k µ w“0 O ǫ , and moreover S` n 3 2 p . ř` q} “ p q | |ě 1´1´ p { q ř 2 Let T S` S´, so that T 3 2 p . The foregoing shows that f iPT xi µ w“0 “ Y | | ď p { q 2 } ´p´ ` q} “ O ǫ and so 1 f xi 1 µ 0 O ǫ . As in Case 1, Lemma 4.10 implies that in p q }p ´ q ´ p iPT ` ´ q}w“ “ p q ř fact T O ?ǫ p . | |“ p { q ř If µ 0 then let g fi 1, and if µ 1 let g fi maxiPT xi. In both cases g is the Boolean rounding “ “ 2 2 of xi 1 µ, and so 1 f g 0 O ǫ . It follows that 1 f g O ǫ , completing iPT ` ´ }p ´ q´ }w“ “ p q }p ´ q´ } “ p q the proof. ř 4.5 Hypergeometric estimate To complete the proof of Theorem 3.1, we present the rather technical proof of Lemma 4.7.

Proof of Lemma 4.7. The distribution of X is given by

t n´t s k´s Pr X s n . r “ s“ ` ˘`k ˘ t ` ˘ ´1 If t 3 then Pr X 0 Ωt 1 p Ωt 1 and t 3 2 2 3 2 p . Similarly, if k 3 then ď r “ s“ k pp ´ q q“ p q ď p { q ď´ p1 { q ď Pr X 0 Ωk 1 t n Ωk 1 and t n 2 3 3 2 p . We can therefore assume that r “ s “ pp ´ { q q “ p q ď { ď ď p { q t, k 4. ě pk`1qpt`1q In view of showing that the mode of X, given by the classical formula s0 t u, is “ n`2 attained at zero, assume that s0 1. Note that s0 2 k since otherwise s0 k 1 and so ě ` ď ě ´ k 1 t 1 k 1 n 2 , implying t 1 3 5 n 2 6 5 t 1 , which is impossible. p ` qp ` q ě p ´ qp ` q ` ě p { qp ` q ě p { qp ` q Similarly, s0 2 t. ` ď

13 PrrX“s`1s pt´sqpk´sq A simple calculation shows that ρs ﬁ . Therefore PrrX“ss “ ps`1qpn´t´k`s`1q

ρs 1 t s 1 k s 1 s 1 n t k s 1 ` ´ ´ ´ ´ ` ´ ´ ` ` ρs “ t s k s s 2 n t k s 2 ´ ´ ` ´ ´ ` ` 1 1 1 1 1 1 1 1 . “ ´ t s ´ k s ´ s 2 ´ n t k s 2 ˆ ´ ˙ˆ ´ ˙ˆ ` ˙ˆ ´ ´ ` ` ˙ Since t s0 2, k s0 2, s0 2 2 and n t k s0 2 2 (using t, k n 2), we conclude ´ ě ´ ě ` ě ´ ´ ` ` ě ď {2 ρs0 that ρs0 1 ρs0 1 16. This shows that Pr X s0 2 ρs0 1ρs0 Pr X s0 Pr X s0 . ` { ě { r “ ` s“ ` r “ sě 16 r “ s If Pr X s0 1 3 then Pr X m,m 1 2 3 and so γ 1 3. We can therefore assume r “ sď { r P t ` us ď2 { ě 2{ that Pr X s0 1 3, and so Pr X s0 2 ρs0 16 1 3 Ω ρs0 . Since m,m 1 cannot r “ sě { r “ ` s ě p 2 { qp { q“ p q t ` u contain both s0 and s0 2, this shows that γ0 Ω ρ , implying that we can assume that ρs0 τ0 ` “ p s0 q ă for some small τ0. pk`1qpt`1q Suppose now that s0 1. Then s0 1, and so ě ą n`2 ´ pk`1qpt`1q pk`1qpt`1q t n`2 k n`2 ρs0 p 1´ 1 qp ´ 1 1q ě pk` qpt` q n t k pk` qpt` q n`2 p ´ ´ ` n`2 q k`1 t`1 t 1 1 n`2 k 1 1 n`2 Ω 1 p ` qp ´ 1 qp1 ` qp ´ q Ω 1 . ě p q pk` qpt` q O n “ p q n`2 ¨ p q By choosing τ0 (and so γ0) appropriately, we can conclude that s0 0, which shows that 1 pk`1qpt`1q 1 “ ą pt, and so t p´ . Moreover, Pr X 0 Pr X s0 1 3. n`2 ą ă r “ s“ r “ sě { References

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