Fourier Concentration from Shrinkage

Abstract

For a class $${\mathcal{F}}$$F of formulas (general de Morgan or read-once de Morgan), the shrinkage exponent$${\Gamma_{\mathcal{F}}}$$ΓF is the parameter measuring the reduction in size of a formula $${F\in\mathcal{F}}$$FźF after $${F}$$F is hit with a random restriction. A Boolean function $${f\colon \{0,1\}^n\to\{1,-1\}}$$f:{0,1}nź{1,-1} is Fourier-concentrated if, when viewed in the Fourier basis, $${f}$$f has most of its total mass on "low-degree" coefficients. We show a direct connection between the two notions by proving that shrinkage implies Fourier concentration: For a shrinkage exponent $${\Gamma_{\mathcal{F}}}$$ΓF, a formula $${F\in\mathcal{F}}$$FźF of size $${s}$$s will have most of its Fourier mass on the coefficients of degree up to about $${s^{1/\Gamma_{\mathcal{F}}}}$$s1/ΓF. More precisely, for a Boolean function $${f\colon\{0,1\}^n\to\{1,-1\}}$$f:{0,1}nź{1,-1} computable by a formula of (large enough) size $${s}$$s and for any parameter $${r > 0}$$r>0, $$\sum_{A\subseteq [n]\; :\; |A|\geq s^{1/\Gamma} \cdot r} \hat{f}(A)^2\leq s\cdot{\mathscr{polylog}}(s)\cdot exp\left(-\frac{r^{\frac{\Gamma}{\Gamma-1}}}{s^{o(1)}} \right),$$źA⊆[n]:|A|źs1/Γ·rf^(A)2źs·polylog(s)·exp-rΓΓ-1so(1),where $${\Gamma}$$Γ is the shrinkage exponent for the corresponding class of formulas: $${\Gamma=2}$$Γ=2 for de Morgan formulas, and $${\Gamma=1/\log_2(\sqrt{5}-1)\approx 3.27}$$Γ=1/log2(5-1)ź3.27 for read-once de Morgan formulas. This Fourier concentration result is optimal, to within the $${o(1)}$$o(1) term in the exponent of $${s}$$s. As a standard application of these Fourier concentration results, we get that subquadratic-size de Morgan formulas have negligible correlation with parity. We also show the tight $${\Theta(s^{1/\Gamma})}$$ź(s1/Γ) bound on the average sensitivity of read-once formulas of size $${s}$$s, which mirrors the known tight bound $${\Theta(\sqrt{s})}$$ź(s) on the average sensitivity of general de Morgan $${s}$$s-size formulas.