Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy

Let Tϵ, 0≤ϵ≤1/2, be the noise operator acting on functions on the boolean cube {0,1}n. Let f be a distribution on {0,1}n and let q>1. We prove tight Mrs. Gerber-type results for the second Rényi entropy of Tϵf which take into account the value of the qth Rényi entropy of f. For a general function f on {0,1}n we prove tight hypercontractive inequalities for the ℓ2 norm of Tϵf which take into account the ratio between ℓq and ℓ1 norms of f.


Introduction
This paper considers the problem of quantifying the decrease in the ℓ 2 norm of a function on the boolean cube when this function is acted on by the noise operator.
Given a noise parameter 0 ≤ ǫ ≤ 1/2, the noise operator T ǫ acts on functions on the boolean cube as follows: for f : {0, 1} n → R, T ǫ f at a point x is the expected value of f at y, where y is a random binary vector whose i th coordinate is x i with probability 1 − ǫ and 1 − x i with probability ǫ, independently for different coordinates.Namely, (T ǫ f ) (x) = y∈{0,1} n ǫ |y−x| (1 − ǫ) n−|y−x| f (y), where | • | denotes the Hamming distance.We will write f ǫ for T ǫ f , for brevity.
Note that f ǫ is a convex combination of shifted copies of f .Hence, the noise operator decreases norms.Recall that the ℓ q norm of a function is given by f q = (E |f | q ) 1 q (the expectations here and below are taken w.r.t. the uniform measure on {0, 1} n ).The norm sequence { f q } q increases with q.An effective way to quantify the decrease of ℓ q norm under noise is given by the hypercontractive inequality [2,5,1] (see also e.g., [6] for background), which upperbounds the ℓ q norm of the noisy version of a function by a smaller norm of the original function.
This inequality is essentially tight in the following sense.For any p < 1 + (q − 1)(1 − 2ǫ) 2 there exists a non-constant function f : {0, 1} n → R with f ǫ q > f p .
Entropy provides another example of a convex homogeneous functional on (nonnegative) functions on the boolean cube.For a nonnegative function f let the entropy of f be given by Ent(f ) = E f log 2 f − E f log 2 E f .The entropy of f is closely related to Shannon's entropy of the corresponding distribution f /Σf on {0, 1} n , and similarly the entropy of f ǫ is related to Shannon's entropy of the output of a binary symmetric channel with error probability ǫ on input distributed according to f /Σf (see below and, e.g., the discussion in the introduction of [13]).The decrease in entropy (or, correspondingly, the increase in Shannon's entropy) after noise is quantified in the "Mrs.Gerber's Lemma" [15]: where ψ = ψ(x, ǫ) is an explicitly given function on [0, 1] × [0, 1/2], which is increasing and strictly concave in its first argument for any 0 < ǫ < 1 2 .This inequality is tight iff f is a product function with equal marginals.That is, there exists a function g : {0, 1} → R, such that for any x = (x 1 , ..., x n ) ∈ {0, 1} n holds f (x) = n i=1 g (x i ).One has ψ(0, ǫ) = 0 and ∂ψ ∂x |x=0 = (1 − 2ǫ) 2 .Hence ψ(x, ǫ) ≤ (1 − 2ǫ) 2 • x, with equality only at x = 0. Hence the inequality (2) has the following weaker linear approximation version which is tight if and only if f is a constant function.
Note that taking q → 1 in this inequality recovers only the (weaker) linear approximation version (3) of Mrs. Gerber's inequality (2).This highlights a important difference between inequalities (1) and (2).Mrs. Gerber's lemma takes into account the distribution of a function, specifically the ratio between its entropy and its ℓ 1 norm.When this ratio is exponentially large in n, which typically holds in the information theory contexts in which this inequality is applied, (2) is significantly stronger than (3).On the other hand, hypercontractive inequalities seem to be typically applied in contexts in which the ratio between different norms of the function is subexponential in n, and there are examples of such functions for which (1) is essentially tight.
With that, there are several recent results [10,8,16] which show that (1) can be strengthened, if the ratio f q f 1 , for some q > 1, is exponentially large in n.In the framework of Rényi entropies, the possibility of a result analogous to (2) for higher Rényi entropies was discussed in [4].
Our results.This paper proves a Mrs. Gerber type result for the second Rényi entropy, and a hypercontractive inequality for the ℓ 2 norm of f ǫ which take into account the ratio between ℓ q and ℓ 1 norms of f .We try to pattern the results below after (2).
We start with a Mrs. Gerber type inequality.
Proposition 1.1: Let q > 1, and let f be a nonnegative function on {0, 1} n such that E f = 1.Then where ψ 2,q is an explicitly given function on [0, 1] × [0, 1/2], which is increasing and concave in its first argument.
This inequality is essentially tight in the following sense.For any 0 < x < 1 and 0 < ǫ < 1 2 , and for any y < ψ 2,q (x, ǫ) there exists a sufficiently large n and a nonnegative function f on {0, 1} n with E f = 1, Let us make two comments about this result.
-The functions {ψ 2,q } q are somewhat cumbersome to describe.Their precise definition will be given below.
-Recall that for a point x ∈ {0, 1} n and 0 ≤ r ≤ n, the Hamming sphere of radius r around x is the set {y ∈ {0, 1} n : |y − x| = r}.As will be seen from the proof of Proposition 1.1, ( 4) is essentially tight for a certain convex combination of the uniform distribution on {0, 1} n and the characteristic function of a Hamming sphere of an appropriate radius (depending on q, ǫ, and the required value of Ent q (f )).
-In information theory one typically considers a slightly different notion of Rényi entropies: For a probability distribution P on Ω, the q th Renyi entropy of P is given by H q (P ) = − 1 q−1 log 2 ω∈Ω P q (ω) .To connect notions, if f is a nonnegative (non-zero) function on {0, 1} n with expectation 1, then P = f 2 n is a probability distribution, and Ent q (f ) = n−H q (P ).Furthermore, Ent q (f ǫ ) = n − H q (X ⊕ Z), where X is a random variable on {0, 1} n distributed accordinng to P and Z is an independent noise vector corresponding to a binary symmetric channel with crossover probability ǫ.Hence, (2) can be restated as and Proposition 1.1 can be restated as Here ϕ is an explicitly given function on [0, 1] × [0, 1/2], which is increasing and convex in its first argument (ϕ(x, ǫ) = 1 − ψ(1 − x, ǫ)), and similarly for ϕ 2,q .
-The precise definition of the functions {κ 2,q } q will be given below.At this point let us just observe that since the sequence {Ent q (f )} q increases with q, we would expect the fact that Ent q (f ) is large to become less significant as q increases.This is expressed in the properties of the functions {κ 2,q } q in the following manner: If q ≥ 2 then for any 0 < ǫ < 1 2 the function κ 2,q (x, ǫ) starts as a constant-1 + (1 − 2ǫ) 2 function up to some x = x(q, ǫ) > 0, and becomes strictly decreasing after that.In other words x(q, ǫ) is the largest possible value of for which Theorem 1.2 provides no new information compared to (1).For 1 < q < 2 there is a value 0 < ǫ(q) < 1 2 , such that for all ǫ ≤ ǫ(q) the function κ 2,q (x, ǫ) is strictly decreasing (in which case we say that x(q, ǫ) = 0).However, x(q, ǫ) > 0 for all ǫ > ǫ(q).The function ǫ(q) decreases with q (in particular, ǫ(q) = 0 for g ≥ 2).The function x(q, ǫ) increases both in q and in ǫ.
-Notably, taking q → 1 in Theorem 1.2 gives (see Corollary 1.5) Hence, this is stronger than (1) for any non-constant function f and for any 0 < ǫ < 1  2 , with the difference between the two inequalities becoming significant when Ent f f 1 /n is bounded away from 0. -As will be seen from the proof of Theorem 1.2, (5) is essentially tight for a certain convex combination of the uniform distribution on {0, 1} n and characteristic functions of one or two Hamming spheres of appropriate radii (the number of the spheres and their radii depend on q, ǫ, and the required value of Ent q f f 1 ).
-Let f be a non-constant function and let 0 < ǫ < 1 2 be fixed.Consider the function , ǫ .It will be seen that there is a unique value 1 < q(f, ǫ) ≤ 1 + (1 − 2ǫ)2 of q for which F (q) = q.Furthermore, q(f, ǫ) = min q≥1 F (q). Hence it provides the best possible value for κ in Theorem 1.2.With that, determining q(f, ǫ) might in principle require knowledge of all the Renyi entropies Ent q (f ), for 1 ≤ q ≤ 1 + (1 − 2ǫ) 2 , while typically we are in possession of one of the "easier" Renyi entropies, such as Ent(f ) or Ent 2 (f ).
Logarithmic Sobolev inequalities.Viewing both sides of (1) as functions of ǫ, and writing L(ǫ) for the LHS and R(ǫ) for the RHS, we have L(0) = R(0) = f 2 , and L(ǫ) ≤ R(ǫ) for 0 ≤ ǫ ≤1 2 .Since both L and R are differentiable in ǫ this implies L ′ (0) ≤ R ′ (0).This inequality is the logarithmic Sobolev inequality for the Hamming cube [5].We proceed to describe it in more detail.Recall that the Dirichlet form E(f, g) for functions f and g on the Hamming cube is defined by E(f, g) = E x y∼x f (x) − f (y) g(x) − g(y) .Here y ∼ x means that x and y differ in precisely one coordinate.The logarithmic Sobolev inequality then states that E(f, f ) ≥ 2 ln 2 • Ent f 2 .Applying the same approach to the inequalities of Theorem 1.2 leads to a family of logarithmic Sobolev inequalities of the form E(f, f ) ≥ c • Ent f 2 , where the constant c depends on Ent q (f ) and belongs to the interval [2 ln 2, 2].Going back to the preceding remark, it is not hard to see that for any function f holds q(f, ǫ) → ǫ→0 2, and hence we take q = 2 in Theorem 1.2 to obtain the following claim.Here and below we write x is a convex and increasing function on The function C was defined in [12], where a somewhat stronger logarithmic Sobolev inequality shown, using a different approach. 1This was used in [12] to establish the following claim, answering a question of [7].Let A ⊆ {0, 1} n .Let M A be the adjacency matrix of the subgraph of the discrete cube induced by the vertices of A. Then the maximal eigenvalue λ(A) of M A satisfies This is almost tight if A is a Hamming ball of exponentially small cardinality.We observe that ( 6) is also an easy implication of Corollary 1.3 (following the argument of [12]), and hence it might be viewed as another consequence of Theorem 1.2.

Full statements of Proposition 1.1 and Theorem 1.2
We now define the functions {ψ 2,q } q in Proposition 1.1 and {κ 2,q } q in Theorem 1.2, completing the statements of these claims.We start with introducing yet another function on [0, 1]×[0, 1/2] which will play a key role in what follows (we remark that this function was studied in [8]).For The function Φ is nonpositive.It is increasing and concave in its first argument.Additional relevant properties of Φ are listed in Lemma 2.3 below.For a fixed ǫ, it will be convenient to write Definition 1.4: We remark that it is not immediately obvious that the functions ψ 2,q and κ 2,q are well-defined.This will be clarified in the proofs of Proposition 1.1 and Theorem 1.2.
We state explicitly two special cases of Theorem 1.2, which seem to be the most relevant for applications.They describe the improvement over (1), given non-trivial information about Ent(f ) and f 2 .
Corollary 1.5: 1. Taking q → 1 in Theorem 1.2 gives: We observe that both Proposition 1.1 and Theorem 1.2 are based on the following claim ([8], Corollary 3.2; this claim also explains the relevance of function Φ).
Theorem 1.6: Let f be a function on {0, 1} n supported on a set of cardinality at most 2 xn .Then, for any 0 Moreover, this is tight, up to a polynomial in n factor, if f is the characteristic function of a Hamming sphere or radius H −1 (x) • n.

Related work
In [10] it was shown that if ǫ) , where ∆(p, ρ, ǫ) > 0 for all p > 1, ǫ, ρ > 0 (cf.with (1), which can be restated as following sense: it is an explicit function of the (unique) solution of a certain explicit differential equation.
In [16] it was shown, using a different approach, that (restating the result in the notation of this paper) f ǫ 2 ≤ f q , where q is determined by F f,ǫ (q) = q (in the notation of the last comment above to Theorem 1.2).As we have observed, this is the best possible value for κ in Theorem 1.2, but it might not be easy to determine explicitly in practice (compare with Corollary 1.5).
This paper is organized as follows.We prove Proposition 1.1 in Section 2 and Theorem 1.2 in Section 3. We prove the remaining claims, including some technical lemmas and claims made above in the comments to the main results, in Section 4.
2 Proof of Proposition 1.1 We first prove (4) and then show it to be tight.We prove (4) in two steps, using Theorem 1.6 to reduce it to a claim about properties of the function φ ǫ , and then proving that claim.
We start with the first step.It follows closely the proof of Theorem 1.8 in [8], and hence will be presented rather briefly, and not in a self-contained manner.Let f be a function on {0, 1} n , for which we want to show (4).We may assume, w.l.o.g., that f is positive, in fact that f ≥ 2 −n .
Furthermore, Theorem 1.6 implies the following fact.We can partition {0, , where v i is the minimal value of f on A i , so that, up to a negligible (vanishing with n → ∞) error holds.
The negligible error we have mentioned contributes towards a negligible error in (4), which can then be removed by a tensorization argument, so we will ignore it from now on.
. Hence, in particular, N ≤ q−1 q .Note also that for any 1 This discussion leads to the definition of the following two subsets of R 2 , which will play an important role in the proof of Theorem 1.2 as well.(We remark that the relevance of the set Ω in the following definition is not immediately obvious.It will be made clear in the following arguments.)Definition 2.1: Let q > 1 and 0 < N ≤ q−1 q .Let Ω 0 ⊆ R 2 be defined by Let Ω ⊆ Ω 0 be the set of all pairs (α, ν) ∈ Ω 0 with ν ≥ 0.
By the preceding discussion, (4) will follow from the following claim.
Before proving Lemma 2.2, we collect the relevant properties of the function φ ǫ in the following lemma.
Lemma 2.3: The function φ ǫ has the following properties.

The function g
This lemma will be proved below.For now we assume its correctness, and proceed with the proof of Lemma 2.2

Proof:
Our first observation is that the maximum of φ ǫ (α) + ν on Ω 0 is located in Ω, since for any point (α, ν) ∈ Ω 0 with ν < 0, the point (α, 0) is in Ω.So we may and will replace Ω 0 with Ω in the following argument.
Since φ ǫ is increasing, any local maximum of φ ǫ (α) + ν is located on the upper boundary of Ω, that is on the piecewise linear curve which starts as the straight line α q + ν = N + 1 q , for 0 ≤ α ≤ 1 − qN q−1 and continues as the straight line α The function h describes the restriction of φ ǫ (α) + ν to the line α q + ν = N + 1 q , and we are interested on the maximum of h on the interval . By Lemma 2.3, the function h is concave, and hence there are two possible cases: In this case h is increasing on I and we get max The last equality follows from the definition of ψ 2,q in this case.
Hence, in this case the maximum of h on I is located at the unique zero of its derivative, that is at the point α 0 such that φ ′ ǫ (α 0 ) = 1 q .Using the definition of ψ 2,q in this case, we get max This concludes the proof of (4).The fact that ψ 2,q (x, ǫ) is strictly increasing and concave in its first argument is an easy implication of Lemma 2.3.
We pass to showing the tightness of (4).Let 0 < ǫ < 1 2 and 0 < x < 1. Set N = q−1 q • x.Let Ω be the domain defined in Definition 2.1, and let (α * , ν * ) be the maximum point of φ ǫ (α) + ν on Ω (note that the discussion above determines this point uniquely).We proceed to define the function f .Let n be sufficiently large.For y ∈ {0, 1} n , let |y| denotes the Hamming weight of y, that is the number of 1-coordinates in y.Let r = ⌊H −1 (α * ) • n⌋.Let S = {y ∈ {0, 1} n , |y| = r} be the Hamming sphere around zero of radius r in {0, 1} n .Now there are two cases to consider.
q , then by the discussion above, the point (α * , ν * ) lies on the line α q + ν = N + 1 q , but not on the line α + ν = 1.Observe that 2 α * n−o(n) ≤ |S| ≤ 2 α * n (the first estimate follows from the Stirling formula, for the second estimate see e.g., Theorem 1.4.5. in [9]).As the first attempt, let g However, E g is exponentially small.To correct that, we define f to be v = 2 (ν * −δ)•n on S, and 2 n −|S|v on the complement of S. Then E f = 1.We choose δ to be as small as possible, while ensuring that Since the contribution of the constant-1 function to f q is exponentially small w.r.t.f q , we can choose δ = o n (1).We now have E f = 1, Here the second equality follows from the semigroup property of the noise operator: . The first inequality follows from the tightness part of Theorem 1.6 and the definition of φ ǫ .The second inequality follows from Lemma 2.2.
The tightness of (4) in this case now follows, taking into account the fact that ψ 2,q is strictly increasing.
q , the point (α * , ν * ) lies on the intersection of the lines α q + ν = N + 1 q , and α + ν = 1.Hence the function g It is easy to see that g can be corrected as in the preceding case, by decreasing it slightly on S and adding a constant component, to obtain a function f with expectation 1 and Ent q (f ) ≤ x, and with Ent 2 (fǫ) n ≥ ψ 2,q (x, ǫ) − o n (1), proving the tightness of (4) in this case as well.We omit the details.

Proof of Theorem 1.2
The high-level outline of the argument in this proof is similar to that of Proposition 1.1.We start with proving (5), doing this in two steps.In the first step Theorem 1.6 is used to reduce (5) to a claim about properties of the function φ ǫ .That claim is proved in the second step.
We will give only a brief description of the first step since, similarly to the first step in the proof of Proposition 1.1, it follows closely the proof of Theorem 1.8 in [8].Let f be a function on {0, 1} n , for which we may and will assume that f ≥ 2 −n and that E f = f 1 = 1.There are O(n) real numbers 0 ≤ α 1 , ..., α r ≤ 1 and −1 ≤ ν 1 , ..., ν r ≤ 1, such that, up to a negligible error, which may be removed by tensorization, we have Hence (5) reduces to claim (7) in the following proposition.
We continue to prove (7), assuming from now on that N > 0 and that N + 1 q > 1 q 0 .Let Ω ⊆ R 2 be the set defined in Definition 2.1.We now define a family of continuous functions on Ω, which will play an important role in the following argument.Let (α 1 , ν 1 ) be a point in Ω with Lemma 3.2: For any choice of (α 1 , ν 1 ) as above the function f α 1 ,ν 1 is well-defined and continuous on Ω.
Let M (α 1 , ν 1 ) = max Ω f α 1 ,ν 1 .The inequality (7) will follow from the next main technical claim, describing the behavior of M (α 1 , ν 1 ), as a function of α 1 and ν 1 .Before stating this claim, let us make some preliminary comments.Note that the points 1 − qN q−1 , qN q−1 and 0, N + 1 q are possible choices for (α 1 , ν 1 ).Note also that α 0 in the third part of the claim is well-defined, by the fourth claim of Lemma 2.3.

If
≤ q, then for any choice of (α 1 , ν 1 ) holds , We will prove Lemma 3.2 and Proposition 3.3 below.For now we assume their validity and complete the proof of Proposition 3.1.
We pass to proving the tightness of ( 7), starting with the case N + 1 q ≤ 1 q 0 .In this case, by definition, κ = q 0 .Let κ < κ be given.Observe that since, by assumption, N > 0, we have Let δ > 0 be sufficiently small (depending on N and κ).Set α 2 = 1 − δ and ν 2 = δ.It is easy to see that α 1 , α 2 and ν 1 , ν 2 satisfy the required constraints.We claim that φ ǫ (α 2 ) + ν 2 > max 1≤i≤2 α i −1 κ + ν i .In fact, for a sufficiently small δ we have, using the second claim of Lemma 2.3 (and observing that φ ′ ǫ is continuous), that and We pass to the case N + 1 q > 1 q 0 and . Set It is easy to see that α 1 , α 2 and ν 1 , ν 2 satisfy the required constraints.It is also easy to see that for any κ < κ holds It remains to deal with the case N + 1 q > 1 q 0 and 1−α is larger than N + 1 q at α = 1 − qN q−1 , and hence the fourth claim of Lemma 2.3 implies that α 2 = α 0 > 1 − qN q−1 .Using this, it is easy to see that α 1 , α 2 and ν 1 , ν 2 satisfy the required constraints.Furthermore, note that α 2 < 1 (again, using the fourth claim of Lemma 2.3).It is also easy to verify, using the definition of α 0 , that which implies that for any κ < κ holds φ ǫ (α 2 ) + ν 2 > max 1≤i≤2 This completes the proof of Proposition 3.1.
The following two claims constitute the main steps of the proof of Proposition 3.3.They describe the respective behavior of maxima points of the first and the second type.Lemma 3.6: Let (α * , ν * ) be a maximum point of f of the first type.Then the following two claims hold.
Lemma 3.7: is the unique maximum point of f .This is a maximum point of the second type.
Let (α * , ν * ) be a maximum point of f of the second type in this case.
Lemmas 3.6 and 3.7 will be proved below.At this point we prove Proposition 3.3 assuming these lemmas hold.
We start with the first claim of Proposition 3.3.Let α 1 = 1 − qN q−1 and ν 1 = qN q−1 .Let f = f α 1 ,ν 1 .By the first claim of Lemma 3.7, we have We pass to the second claim of the proposition.Assume that for some α 1 and ν 1 .Let (α * , ν * ) be a maximum point of f .Then Lemmas 3.6 and 3.7 imply that α * ≤ 1 − qN q−1 .Hence Here in the second step we have used the third claim of Corollary 3.5, in the third step the third claim of Lemma 2.3 and in the fourth step the first claim of the proposition.
We pass to the third claim of the proposition.Assume that some α 1 and ν 1 .Then, by Lemma 3.7, f has a unique maximum point (α * , ν * ).This means that α * is determined by α 1 and ν 1 , and furthermore, since ν 1 = N + 1−α 1 q , α * is a function of α 1 .We will show the following claim below.
To conclude the proof of the third claim of the proposition, observe that since α * > 1 − qN q−1 , we have where the last inequality is by the third claim of Lemma 2.3.This completes the proof of Proposition 3.3.
We can now complete the proof of the lemma.First, let (α 1 , ν 1 ) = 1 − qN q−1 , qN q−1 .By the preceding discussion, in this case a maximum point (α * , ν * ) of f of the second type has to have α * ≤ α 1 .Moreover, taking into account Lemma 3.6, this is true for any maximum point of f .By the third claim of Corollary 3.5, this means that M (α 1 , ν 1 ) ≤ α 1 −1 φǫ(α 1 ) = f (α 1 , ν 1 ).Hence (α 1 , ν 1 ) is a maximum point of f .It is trivially a maximum point of the second type.To see that it is a unique maximum point, note that for any point (α, ν) on the upper boundary of Ω, if α = α 1 , then necessarily ν = ν 1 .So, for any other putative maximum point (α, ν), we would have α < α 1 and hence, by the third claims of Lemma 2.3 and the third claim of Corollary 3.5, . This proves the first claim of the lemma.
In the assumptions of the lemma, α * is the unique solution on 1 − qN q−1 , 1 of the identity .
We pass to the last claim of the lemma.Taking the derivative and rearranging, it suffices to prove that for any α ∈ (0, 1) holds , and this follows from the first two claims of the lemma.The values of the function g at the endpoints are directly computable.
The first claim of the lemma is verified by inspection, observing that y(x, 0) = 0 for any 0 ≤ x ≤ 1.
We pass to the second claim of the lemma.Using (as in the proof of Lemma 2.13 in [8]), the fact that for ǫ > 0 holds (σ−y)(1−σ−y) • lim .
This proves the second claim of the lemma.
In the following we will assume (as we may, since for constant functions the claim of the corollary is trivially true) that f is not a constant function, implying that x = 1 n Ent 2 f f 1 > 0. For ǫ sufficiently close to zero, we have that x+1 2 > 1 q 0 (recall that q 0 = 1 + (1 − 2ǫ) 2 ) and hence κ = .
Proof of Corollary 1.5 Proof: We start with the first claim of the corollary.First consider the case ǫ = 1 2 .It is easy to see that φ 1 2 (x) = x − 1 (note that in the definition of Φ(x, ǫ) we have y x, 1 2 = lim ǫ→ 1 2 y (x, ǫ) = H −1 (x) 1 − H −1 (x) ) and hence in this case the value of κ given by the claim is 1 (as it should be).
We pass to the second claim of the corollary.First consider the case ǫ = 0. Note that in this case q 0 = 2. Furthermore, by the first claim of Lemma 4.1, φ 0 (x) = x−1 2 , and hence the value of κ given by the claim is 2 (as expected).
Assume now ǫ > 0. This implies that q 0 < 2, and hence, by the third claim of Lemma 2.3, for any 0 ≤ x ≤ 1 we have −x φǫ(1−x) ≤ q 0 < 2 = q.Hence the second clause in the definition of κ 2,q in Definition 1.4 does not apply.The remaining two clauses give the claim, as stated.

log 2 1 1
−t for the binary entropy function.Corollary 1.3:For any function f on {0, 1} n holds