Quasi-Random Multilinear Polynomials 3

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SCHULMAN below. The upper bound 2(n + 1)u is also straightforward; generally it is weaker than the bound ε controlled by √ui, sometimesp substantially so (e.g., consider the disjoint union of a clique on n vertices, ε > P1/2, with a matching on n nε vertices). There are situations where it is stronger, − especially when d is large and the ui are balanced, but it cannot be stronger by more than a factor of 3.34 d/2, due to Cauchy-Schwarz: √u n u √dnu. i ≤ i ≤ It isp not really necessary in the theoremP to assumep UPis multilinear, since exponents may be reduced by 2 without affecting evaluation on ( 1)n; in the process however monomials identify and their coefficients add. One can write down± bounds in terms of general coefficients but for brevity and because of connections we sketch shortly, we restrict our statements to Littlewood polynomials. It is interesting to also study inf h(Dn) . h U,C k k∞ ∈L n By the maximum modulus principle, for any U (not necessarily multilinear) and h U,C, h(C ) = n ∈L k n k∞ h(D ) . Our upper and lower bounds can be replicated with mild loss for infh U,C h(D ) ; wek omitk doing∞ so since no new ideas are involved. The upper bound does not require∈L thek multilineark∞ assumption.

2. BACKGROUND AND RELATED LITERATURE

S 2.1. An elementary lower bound. For a polynomial h = S hSx (not necessarily multilinear, so S may range over multisets), and for 1 α< , let P ≤ ∞ 1/α n n α h(( 1) ) α = 2− h(x)  k ± k | | x X( 1)n  ∈ ±  and 2π 2π 1/α n n iθ1 iθn α h(C ) α = (2π)− h(e ,...,e ) dθ1 dθn . k k Z0 · · · Z0 | | · · · These are the evaluation norms of the polynomial (normalized to be means), with α = being the α 1/α ∞ max norm. The coefficient norms are h α = ( S hS ) with of course h = maxS hS k k | | k k∞ | | (these are not normalized to be means). The power-meansP inequality gives n n (2.1) h(C ) h(C ) 2 k k∞ ≥ k k and considering C as the circle group, we have the Plancherel identity (2.2) h(Cn) = h = u1/2 k k2 k k2 Let L(d) denote the complex univariate polynomials of degree d, all of whose coefficients are units; these are customarily called Littlewood polynomials (we have adopted the term more broadly). In 1957 Erdös [16] asked for the minimum over L(d) of the sup norm of polynomials on the unit disk D, i.e., for minh L(d) maxx D h(x) ; he also investigated the analogous question for trigonometric ∈ ∈ | | polynomials [17]. Littlewood [31, 32] asked for minh L(d)((maxx C h(x) ) (minx C h(x) )); this is known as Littlewood’s flatness problem. The question∈ s have∈ been| investigated| − for∈ both| com-| plex and real (i.e., 1) coefficients. Our paper concerns Erdös’s question as applied to multilinear polynomials. ±

For univariate Littlewood polynomials of degree d, (2.1)-(2.2) specializes to h(C) h(C) 2 = k k∞ ≥ k k h 1 h . The last two terms are necessarily equal, and equal to √d + 1. There has been k k2 ≥ √d+1 k k1 careful study of how tight this inequality is. First, for the case of real (i.e., 1) coefﬁcients, the ± QUASI-RANDOM MULTILINEAR POLYNOMIALS 3

Rudin-Shapiro polynomials [45, 43, 31] match the lower bound to within a factor of √2. It is an open problem to close this gap. For the case of complex coefficients, the gap to the lower bound is (1+ εn) for some εn 0 (these are called “ultraflat” polynomials); this construction is due to Kahane [24] (and see a correction→ in [40]). For a survey of these and considerably more results in the univariate case, see [15, 3]; for some recent results see [22, 44]. A bound analogous to (2.1)-(2.2) holds with the group Z/2 replacing the circle group: Let U be multilinear. For h R, ∈LU, n n 1/2 (2.3) h(( 1) ) h(( 1) ) 2 = h 2 = u . k ± k∞ ≥ k ± k k k (This establishes the simpler of the two lower bounds in (1.2).) The inequality is again power means. The equality can be understood in two equivalent ways. The first is that each multilinear monomial xS is a multiplicative character on the group (Z/2)n and so the transformation between the vector of coefficients h and the vector of evaluations h(( 1)n) is the Fourier transform over (Z/2)n, so this is S ± the Parseval identity. The second way is to consider each xi to be an iid uniform random variable on 1. Then the monomials xS are also uniform on 1, and, since they are multilinear, they are pairwise independent.± Since E(h(x)) = 0, h(( 1)n) 2±= E(h2(x)) = Var(h(x)) = Var(h xS) = k ± k2 S S h2 = h 2. P S S k k2 P 2.2. Quasi-random graphs and hypergraphs. Chung, Graham and Wilson [10] (and for precursors see [49, 50, 41, 19]) define a family of graphs Gi = (Vi, Ei) to be quasi-random if, among several equivalent characterizations, for any functions x : V 1, i i →±

x (u)x (v) x (u)x (v) o( V 2). i i − i i ∈ | i| u,vXE u,vX/E { }∈ i { }∈ i

Simply by setting huv = 1 for edges and huv = 1 for non-edges, this expression exactly corresponds − Vi to evaluation of a multilinear Littlewood polynomial with U = 2 . A quasi-random graph is one whose corresponding polynomial achieves a non-trivial, that is sub-quadratic (though not necessarily optimal in the exponent) sup-norm bound.

Let Ai be the adjacency matrix of Gi (and note the graph has no loops or multiple edges); and consider x as a column vector (thus x = V ). Then an equivalent characterization of quasi-random k k2 | i| graphs is that p 2 x∗(2A + I J)x / x o( V ) | i − | k k2 ∈ | i| with I the identity matrix and J the all-ones matrix. In fact, an example given in [10] yields a deterministically constructible polynomial of degree 2 with an optimal (up to the constant) norm bound. Speciﬁcally, the Paley graph Pn (vertices are elements of the ﬁeld GF (n) for n = pα, p prime 1 mod 4; i, j is an edge if i j is a square) gives a ≡ n { } 1 − homogeneous degree 2 polynomial hPn with u = 2 such that n 3/2 hPn (( 1) ) n . k ± k∞ ≤ n 3/2 Theorem 1 establishes on the other hand that hPn (( 1) ) Ω(n ). k ± k∞ ∈ 1 The Paley graph is strongly regular [28] and its adjacency matrix has the spectrum: λ1 =(n 1)/2 with multiplicity − one (eigenvector v1 = the constant function), and λ2,...,λn ( 1 √n)/2 (each with multiplicity (n 1)/2). Then ∈ − ± − with A the adjacency matrix of Pn, (2A + I J) commutes with A and has spectrum 0 for v1, 2λi + 1 for remaining − n ∗ 2 3/2 eigenvalues, so by Courant-Fischer, for x ( 1) , x (2A + I J)x x 2 maxi≥2 2λi + 1 = n . ∈ ± | − | ≤ k k | | 4 GIL KALAI AND LEONARD J. SCHULMAN

This connection between graphs and polynomials is the reason we adopt the term “quasi-random” for a polynomial which achieves a sup-norm which is o(u). However this is a fairly soft requirement, and as indicated in Theorem 1, a random polynomial achieves a much tighter bound. We mention also that Simonovits and Sós [46] established another characterization of quasi-random graphs: they are those which, in the partitions provided by the Szemerédi regularity lemma, have almost all inter-block edge densities close to 1/2. Through this connection quasi-random graphs play a role in the theory of graph limits; see Lovász’s monograph [33]. Similar quasi-randomness questions have also been explored for hypergraphs [21, 9, 8, 26, 11]. The last of these defines property DISCr(δ) for a d-uniform hypergraph to be that as n and for S d → ∞ all S [n], the number of induced d-hyperedges in S is in the interval r | | δn ; a hypergraph ⊆ d ± achieving small δ (i.e., having a number of induced hyperedges that is closely predicted just by S ) is to be considered quasi-random. We can again convert a d-uniform hypergraph to a multilinear| | polynomial by putting, for all S [n] , h = 1 if S is in the hypergraph and h = 1 otherwise. ∈ d S S − However, for d > 2, there does not seem to be a close connection between being quasi-random in the sense of DISC and being quasi-random in the sense of the sup-norm of the polynomial. DISC essentially counts how many hyperedges intersect but are not supported on S; it does not distinguish further among these hyperedges based on the cardinality of their intersection with S. However, a different single-number summary of the intersection frequencies starts with the perspec- tive that graph expansion measures the quality of the graph as a cut approximator of the complete graph, which is the same thing as how well it approximates the second binary Krawtchouk polynomial [27, 30] at 1/2; equivalently, edges are counted according to the parity of their intersection with [n] S. The polynomial sup-norm that we study generalizes this for U = d : it measures the quality of a d-uniform hypergraph as a cut approximator of the d’th binary Krawtchouk polynomial at 1/2. Equivalently, hyperedges are counted according to the parity of their intersection with S. This same quantity is the key measure of hypergraph expansion in the work of Ta-Shma on small ε-biased sets [47].

2.3. Spin glasses. In statistical mechanics and condensed matter, physicists have long studied “local Hamiltonians”, that is, energy functions deﬁned in terms of local interactions between particles. Locality usually takes its meaning from a graph embeddable without distortion in one to three dimensions of space (often but not always a lattice). The Ising model (see, e.g., [39]) is a common model in which each particle takes on just two states 1, and, with V denoting the set of particles and U the edges between them, the energy of the± system in state x : V 1 is → ± H(x) = u,v U h u,v x(u)x(v). A ferromagnetic Ising model is one in which h u,v 0 − { } { } ≥ for all u, v P{U;}∈ this is a model whose ground state is frustration-free, namely, all contributions to the energy{ are}∈ simultaneously at their least possible value. In our setting this “ferromagnetic polynomial” h achieves the worst-possible (largest) norm bound. On the other hand spin glasses are models h which necessarily exhibit a great deal of cancelation among the contributions to the energy; in our setting this is a polynomial with a strong norm bound. Typically in physics one considers a random h to be a spin glass. Our work shows that such systems can achieve non-trivially low energies, i.e., signiﬁcantly less “frustration” than one would see in a random state x.

2.4. Bent functions. Bent functions [42, 12, 13] are functions ξ : 0, 1 n 0, 1 such that the function ( 1)ξ, which is of course perfectly “flat” (all values have a{ common} →norm), { } has a Fourier − n n/2 transform over (Z/2) that is also perfectly flat, namely (with the transform scaled by 2− ) the transform is ( 1)η for another function η : 0, 1 n 0, 1 . Thus each of ( 1)ξ and ( 1)η is − { } → { } − − QUASI-RANDOM MULTILINEAR POLYNOMIALS 5 a perfectly flat multilinear Littlewood function; as we have seen, this is possible only due to their high degree. Bent functions have been extensively studied particularly in view of applications in cryptography [51, 6] and quantum computation [7].

3. PROOF OF THEOREM 1: UPPERBOUND

The case d = 1 is trivial; now assume d 2. We employ the generic chaining following Tala- ≥ grand [48]. Order the variables so that u1 . . . un 1. Set K = lg(n + 1) and partition U ≥ ≥ k ≥1 k ⌈ k⌉ into the sets Vk, 1 k K, Vk = S U : S [2 − , 2 1] = and S [2 ,n] = . Then min≤2k ≤1,n { ∈ ∩ − 6 ∅ ∩ ∅} v := V { 1− } u . Note that v 1. k | k|≤ i=2k− i 1 ≤ P k/2 1/2 Lemma 2. There is an h s.t. maxx ( 1)n h(x) 1.18 k 1 2 vk . ∈ ± | |≤ P ≥ n Proof. For x ( 1) let x(k) = (x1,...,x2k 1, 0,..., 0) (with x(0) = (0,..., 0) and x(K)= x). ∈ ± − It is useful to expand h(x)= K h(x(k)) h(x(k 1)). For k = 1,...,K, in order, we will fix k=1 − − the coefficients h : S V Pso that for all x, h(x(k)) h(x(k 1)) 1.18 2k/2v1/2. { S ∈ k} | − − |≤ · k Suppose that hS has been fixed for all S V1 . . . Vk 1. Choosing hS uniform iid for all S Vk, h(x(k)) h(x(k 1)) is a symmetric random∈ ∪ walk∪ of length− v . By a standard Chernoff bound∈ [35], − − k for any λ> 0, Pr[ h(x(k)) h(x(k 1)) > λ2k/2v1/2] 2 exp( λ22k 1). | − − | k ≤ − − Taking a union bound over x(k), we have k/2 1/2 2k 2 k 1 (1 λ2(lg e)/2)2k Pr[ x(k) : h(x(k)) h(x(k 1)) > λ2 v ] 2 exp( λ 2 − ) = 2 − . ∃ | − − | k ≤ − In order that there exist a satisfactory choice of coefficients h for S V it suffices that λ > S ∈ k 2/ lg e; e.g., we may take λ = 1.18. p To apply this in the theorem, consider any 1 < k K. Then ≤ 1/2 min(2k 1,n) − k/2 1/2 k/2 2 v 2  ui k ≤ i=2Xk−1   k/2 k 1 1/2 2 (2 − u k−1 ) ≤ 2 k 1/2 1/2 2 − u2k−1 1 ≤ − 2k−1 1 − 23/2 u1/2. ≤ i i=2Xk−2 k/2 1/2 3/2 Also, for k = 1 observe that 2 v √2 1 √2u K−1 2 u K−1 . k ≤ · ≤ 2 ≤ 2 K−1 Thus 1.18 K 2k/2v1/2 1.18 23/2 2 u1/2 3.34 n u1/2. k=1 k ≤ · i=1 i ≤ i=1 i P P P 4. PROOF OF THEOREM 1: LOWERBOUND

The simple bound of u1/2 was shown in (2.3). The case d = 1 is trivial; now assume d 2. In a high-level view, the proof strategy is to reduce to the linear case, in a manner modeled after≥ the solution of the Gale-Berlekamp game [4, 20, 18], which concerns the case of a “bipartite” quadratic polynomial. In our situation, several additional steps are needed: one being to ﬁnd a suitable partition of the variables that replaces this bipartite structure, a second involving a generalization of 6 GIL KALAI AND LEONARD J. SCHULMAN the Khintchine-Kahane inequality, and the last being to use a bound from approximation theory to overcome cancelation by nonlinear terms (this will be clearer below). Our ﬁrst step is to split the variables into two sets X and Y , with a desirable property on the degrees (in the hypergraph of monomials) of the variables in X. 1 Consider selecting a set X as follows: each i is included in X independently with probability 2(d 1) . Let M = a U : a X = i , and m = M . Note that m = 0 for i / X. − i { ∈ ∩ { }} i | i| i ∈ a 1 Conditioned on i X a, Pr(a Mi) 1 2(| d|−1) (applying independence and a union bound). ∈ ∩ ∈ ≥ − ui− Thus (without conditioning), for any i, Emi 4(d 1) . ≥ − ui ui ui ui Since mi ui, we can write 4(d 1) (1 Pr(mi > 8(d 1) )) 8(d 1) + Pr(mi > 8(d 1) )ui, ≤ − ≤ − − − − ui 1 1/2 5/2 3/2 1/2 consequently Pr(mi > 8(d 1) ) 8(d 1) . So for any i, Emi 2− (d 1)− ui . − ≥ − ≥ − Consequently 1 (4.1) E m1/2 u1/2. i ≥ 25/2(d 1)3/2 i X − X Fix a set X achieving (4.1). We proceed to show the existence of an assignment to the variables achieving large h(x) . | | Let Y = [n] X and write h = d h where − s=0 s P S T hs = hS,T x y S X,T Y,SXT U, S =s ⊆ ⊆ ∪ ∈ | | S and where x = i S xi, etc. Q ∈ Decompose h1 into h1 = i X xihi where ∈ ∗ P T hi = h i ,T y . ∗ { } T Y :XT i U ⊆ ∪{ }∈ hi is degree (d 1)-bounded, and hi 1 = mi. Now choose each yi 1 randomly, uniformly and∗ independently.− We show k ∗k ∈ ± 1 d 1/2 (4.2) E hi 3 − m . | ∗|≥ i as an instance of the following Khintchine-type bound [25, 23, 29]: Proposition 3. Let p be a multilinear Littlewood polynomial with w monomials of degrees δ, in variables y which are iid uniform in 1. Then E p 3 δ(Ep2)1/2 = 3 δw1/2. ≤ { j} ± | |≥ − − Conversely there is a p s.t. E p 2(1 δ)/2w1/2. | |≤ −

To obtain (4.2) apply this upper bound to the polynomial p = hi , with coefﬁcients pT = h i ,T1 , on the variables in Y . ∗ { }

Proof. Note

2 T1+T2 2 Ep = pT1 pT2 Ey = pT1 pT2 = pT = w TX1,T2 T1,T2:XT1 T2=0¯ XT ⊕ where + in the exponent is multiset addition (i.e., repetitions are retained), while is multiset addition modulo 2 (and 0¯ denotes multisets with all repetition numbers even). This⊕ is because any monomial with an odd repetition number contributes 0 to the expectation. QUASI-RANDOM MULTILINEAR POLYNOMIALS 7

Next we require an upper bound on Ep4; w3 is obvious but not useful. However, application of the Bonami Lemma [2, 37] gives that (Ep4)1/4 3δ/2(Ep2)1/2, and consequently ≤ (4.3) Ep4 9δ(Ep2)2 = 9δw2. ≤ At this point, for a weaker bound, we could simply apply the Paley-Zygmund inequality [38], which in this setting provides Pr( p > 1 (Ep2)1/2) 9 (Ep2)2/(Ep4) 9 9 δ. Thus E p > 1 91 δw1/2. | | 2 ≥ 16 ≥ 16 − | | 32 − Instead, to complete the proof of Prop. 3 we rely on the idea of Berger [1] to use the inequality: 33/2 x,a > 0 : x (x2 x4/a2) ∀ ≥ 2a − Applying this inequality with x = p , and using (4.3), we ﬁnd | | 33/2 E p E(p2 p4/a2) | | ≥ 2a − 33/2 w w29δ/a2 ≥ 2a − Setting a = 3δ+1/2w1/2 we have δ 1/2 E p 3− w | | ≥ as desired. For the converse, consider the following polynomial (suggested from an example in [34]) in variables xi and yi (1 i δ): ≤ ≤ δ δ p = (1 + x ) (1 + y ) i − i Yi=1 Yi=1 This is a constant-free Littlewood polynomial of degree δ with w = 2δ+1 2 monomials. The only values p can achieve are 0, 2δ and 2δ, the latter values occurring with probabilities− 2 δ(1 2 δ). − − − − So E p = 2 21 δ, and consequently E p 2(1 δ)/2w1/2. | | − − | |≤ − It is an interesting question to resolve the gap between 3 and √2 in the base of this Khintchine bound. Now we prove the lower bound of Theorem 1. Apply (4.2) and linearity of expectation to pick 1 d 1/2 1 d 5/2 the yi’s so that hi 3 − mi ; apply (4.1) to conclude that i hi 3 − 2− (d ∗ ∗ 3/2 1/2 | | ≥ | | ≥ − 1)− ui . ThenP pick the xi’sP so that h1 = i hi . P | ∗| HavingP set the yi’s randomly and tailored the xPi’s to control h1, we have lost all control over the values of h , h ,...,h , each of which lies in the range u. In particular these terms can easily 0 2 d ± cancel out the contribution from h1.

We now show how to remedy the potential cancelation of the h1 term by the remaining terms. We do this by changing the assignment to the xi’s. (This step of the proof is somewhat unusual, but it is not completely new: we are aware that the same idea was used for a different purpose in [14].) The modiﬁcation of the xi’s is probabilistic. We use approximation theory to show that there exists a value of 0 p 1 s.t. if we ﬂip each x independently with probability p, then E h is large. ≤ ≤ i | | Set 1 z = 1 2p 1. Let x (1 i n ) be independent random variables with Pr(x = − ≤ − ≤ i′ ≤ ≤ 1 i′ 1) = 1 p, Pr(xi′ = 1) = p. Write xx′ for the list of products (x1x1′ ,...,xn1 xn′ 1 ). Let f(z) be the polynomial− − d d s s f(z)= E ′ h(xx′,y)= (1 2p) h (x,y)= z h (x,y) x − s s Xs=0 Xs=0 8 GIL KALAI AND LEONARD J. SCHULMAN

What we know about this polynomial is only that the linear coefficient of z, h1(x,y), is large, specif- ically h (x,y) 31 d2 5/2(d 1) 3/2 u1/2. It is useful to note that h (x,y) = f (0). As 1 ≥ − − − − i 1 ′ noted, we have no control over the coefficientsP of the remaining terms in f. But the lower bound on f ′(0) is enough information to guarantee that the polynomial is large somewhere in [ 1, 1]. This is a consequence of a classic result in uniform approximation theory (of which we quote− only a special case): Lemma 4 (Bernstein [36]). Let f be a polynomial of degree d with complex coefficients. Then supz [ 1,1] f(z) f ′(0) /d. ∈ − | | ≥ | |

As a result we immediately obtain that there is a setting of the xi’s such that 1 d 5/2 3/2 1 1/2 h(x,y) 3 − 2− (d 1)− d− ui . | |≥ − X 5. DISCUSSION

5.1. How quasi-random is the determinant? Let U be the collection of monomials that occur as permutation matrices in a d d matrix. Here n = d2 and u = d!. × The determinant is in U,R and the upper bound for the determinant is the volume bound of Hadamard: 1 L d/2 d log d since each row of the matrix has L2 norm at most √d, the volume is at most d = e 2 . This qualifies the determinant as quasi-random by our soft definition, since the trivial upper bound d log d d+O(log d) d/2 is d! e − . More significantly for our current purpose, the d bound is tight for infinitely∈ many d (powers of 2 and also some other values). However, using the (easy) upper bound of Theorem 1, we have that there exists a Littlewood polyno- 1 2 2 (d log d d)+O(log d) mial h with the same monomials such that maxx ( 1)n h(x) 2(d + 1)d! e − . ∈ ± | |≤ ≤ d/2 p This improves on the determinant by a factor of ∼= e . (The simple lower bound for this U is √u = √d!. The upper and lower bounds are within a factor of merely 2(d2 + 1).) In short the determinant is fairly good as a quasi-random polynomial,p but not competitive with a random polynomial supported on the same monomials. We remark that matrices with 1 entries are a subject of extensive study in mathematical physics and beyond, starting with Wigner± [52]; however this literature is apparently not too connected to our question, as the focus is on random (symmetric) matrices (not extremal ones) and the entire spectrum, not just the determinant, is of essence.

n 5.2. infh U,R h(D ) : complex derandomization. For any Littlewood family U, our lower ∈L k k∞n n bound on infh U,R h(I ) is of course also a lower bound on infh U,R h(D ) . Our approximation theory∈L argumentk cank∞ be derandomized in a slightly unusual∈L way.k That is,k∞ applying Bern- stein’s lemma and the maximum modulus principle, we conclude that there is a z C such that ∈ h(zx1,...,zxn1 ,y1,...,yn2 ) achieves the lower bound. Of course we do not know this number z explicitly (any more than in the argument of Section 4 we know p and all the new values of the xi’s).

5.3. Large d; Khintchine; Anti-concentration. Our lower bound in Theorem 1 trivializes at d logarithmic in the size of the hypergraph. The exponential loss occurs in the Khintchine-type inequality, Prop. 3. As noted in the converse portion of that proposition, the exponential loss is unavoidable. The converse also has an implication for the limits of anti-concentration results. An anti-concentration bound on a mean-0 random variable is an upper bound on Pr( p < ε(Ep2)1/2) for some ε; earlier | | QUASI-RANDOM MULTILINEAR POLYNOMIALS 9 we applied Paley-Zygmund in this way for ε = 1/2 but one may seek better bounds for smaller ε, and considerable work has been done on anti-concentration for polynomials. However even in the most tractable case, when the underlying random variables are iid normal [5], bounds for polynomials of degree δ are useful only for Pr( p < (Ep2)1/2/δδ), i.e., for exponentially small ε as a function of δ. These bounds have been extended| | with slight loss to hypercontractive random variables (which includes the 1 case we study), but due to the exponential dependence on δ, such bounds cannot lead to a version± of Theorem 1 that does not suffer the exponential in d. And of course since anti- concentration implies a lower bound on E p , the impossibility of a Khintchine bound sub-exponential in δ also implies that strong anti-concentration| | results cannot be sought for δ more than logarithmic in w. Indeed the p provided as an example in Prop. 3 is a constant-free Littlewood polynomial of degree δ with 2δ+1 2 monomials, for which Pr(p = 0) is close to 1 (speciﬁcally > 1 21 δ). − − − However, despite the Khintchine converse, it is an open question whether the exponential dependence on d in Theorem 1 is necessary.

ACKNOWLEDGMENTS

Part of this work was done while the authors were in residence at the Simons Institute for Theory of Computing, and part while the second author was in residence at the Israel Institute for Advanced Studies, supported by a EURIAS Senior Fellowship co-funded by the Marie Skłodowska-Curie Ac- tions under the 7th Framework Programme. Work of the second author was also supported by United States NSF grant 1618795. Thanks to an anonymous referee for insightful comments which helped tighten our bounds.

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