Fourier growth of degree $2$ polynomials

We prove bounds for the absolute sum of all level-$k$ Fourier coefficients for $(-1)^{p(x)}$, where polynomial $p:\mathbf{F}_2^n \to \mathbf{F}_2$ is of degree $1$ or degree $2$.

💡 Research Summary

The paper studies the Fourier–Walsh expansion of Boolean functions of the form f(x)=(-1)^{p(x)} where p: 𝔽₂ⁿ→𝔽₂ is a low‑degree polynomial. For a function f, the k‑th Fourier weight is defined as W_k(f)=∑_{|A|=k}|ĥf(A)|, i.e., the sum of absolute values of all Fourier coefficients on level k. A conjecture from Chattopadhyay, Hatami, Hosseini and Lovett (CHHL18) predicts that for any degree‑d polynomial p there exists a constant c_d (depending only on d) such that W_k((-1)^{p}) ≤ c_d·k^{d} for all k≥1, and that one can take c_d = 2^{O(d)}. While the linear case (degree 1) is trivial, the quadratic case (degree 2) remained open.

The authors resolve the quadratic case completely. Their first main result (Theorem 1) states that for every polynomial p of degree at most 2 and every k, the Fourier weight satisfies
 W_k((-1)^{p}) ≤ (1+√2)^{k}.
The proof proceeds by a double induction on the number of variables n and on the level k. If p is affine, the bound is immediate because the Fourier spectrum is supported on a single level. For a genuinely quadratic p, after possibly permuting variables, the authors write
 p(x)=p₀(¯x)+x₁p₁(¯x)+x₂p₂(¯x)+x₁x₂,
where ¯x denotes the remaining n‑2 variables. Splitting the sum defining W_k into four parts according to whether the set A contains none, one, or both of the first two variables, they obtain four expressions I₀, I₁, I₂, I₁₂. A simple algebraic identity (Lemma 3)
 1+(-1)^{a}+(-1)^{b}-(-1)^{a+b}=2·(-1)^{ab}
allows each term to be rewritten as a Fourier weight of a quadratic polynomial in n‑2 variables, possibly on a lower level (k, k‑1, or k‑2). Consequently they derive the recurrence
 W(2,k,n) ≤ ½·W(2,k,n‑2) + W(2,k‑1,n‑2) + ½·W(2,k‑2,n‑2).
Applying the induction hypothesis that W(2,·,·) ≤ (1+√2)^{·} to the right‑hand side yields exactly the claimed bound (1+√2)^{k}.

The second main result (Theorem 2) refines the bound to a near‑optimal asymptotic form:
 W_k((-1)^{p}) ≤ √(2/π)·k^{-1/2}·(1+√2)^{k}.
To obtain this sharper estimate the authors use a structural theorem of Dickson (as presented in McWilliams–Sloane) which says that any quadratic Boolean polynomial can be transformed by a nonsingular linear change of variables into a sum of independent products of paired variables plus an affine part:
 q∘T^{-1}(y)=∑{j=1}^{m} y{2j-1}y_{2j} + ℓ·y + b.
Because the Fourier transform is invariant under such linear changes, the Fourier coefficients of f=(-1)^{q} are non‑zero only on subsets that lie in the span of the paired coordinates and possibly the linear term ℓ. This reduces the problem to counting how many weight‑k vectors lie in an affine subspace of dimension 2m. Lemma 5 shows that an affine subspace of dimension h contains at most C(h+1,k) vectors of Hamming weight k. Consequently,
 W_k ≤ 2^{-m}·C(2m+1,k).
The authors then bound the binomial coefficient using Stirling’s approximation and a technical inequality (Lemma 6) to obtain the factor √(2/π)·k^{-1/2}. Lemma 7 provides the precise inequality
 C(2m+1,k) ≤ √(2/π)·k^{-1/2}·(1+√2)^{k},
which is tight: for the quadratic form q(x)=∑{j=1}^{m} x{2j-1}x_{2j} and for k≈(2-√2)m, the ratio of the left‑hand side to the right‑hand side approaches 1. Thus Theorem 2 matches the true maximal Fourier weight up to lower‑order factors, confirming the conjectured exponential dependence on k with the optimal base 1+√2.

Section 3 also contains a discussion of the sharpness of the bounds, showing that the constants cannot be improved asymptotically. The authors remark that the same technique yields the exact constant for the linear case (c₁=1) and that the quadratic case is now completely settled.

In Section 4 the authors sketch how one might extend the analysis to degree 3 polynomials. A cubic polynomial can be written as a sum of a pure cubic term x₁x₂x₃ plus various linear, quadratic, and mixed terms. Decomposing the Fourier weight into eight contributions (depending on which of the first three variables belong to the set A) leads to a more intricate recurrence, but the authors do not yet obtain a clean closed‑form bound. They leave the cubic (and higher‑degree) case as an open problem, suggesting that the methods developed for quadratics may serve as a foundation for future work.

Overall, the paper delivers two independent proofs that fully resolve the Fourier‑weight conjecture for degree‑2 Boolean polynomials: an elementary induction that yields a clean exponential bound (1+√2)^{k}, and a more refined argument based on Dickson’s normal form that attains the optimal constant √(2/π)·k^{-1/2}. The results close a gap in the literature on pseudorandom generator constructions that rely on Fourier‑weight estimates, and they open the door to tackling higher‑degree cases.