An $m^{2.943}$ Bohnenblust--Hille Bound on the Boolean Cube

Abstract

Let $q_m=2m/(m+1)$ and put \[ β_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\ldots, \] where $\log$ is the natural logarithm. We give a proof scheme showing that, for every $\epsilon >0$, there is $C_{\epsilon }<\infty$ such that every complex-valued function $f:\{-1,1\}^n\to\C$ of Fourier degree at most $m$ satisfies \[ \left(\sum_{A\subseteq[n]}\abs{\wh f(A)}^{q_m}\right)^{1/q_m} \le C_\varepsilon m^{β_0+\varepsilon}\norm{f}_\infty. \] The improvement over the $m^9$ estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales $m$, $m^{3/2}$, and $m^{5/3}$. These losses are encoded in the weight $M^{{\mu }_r/r}{r}^B\min \{r,L_M{\}}^8$, $L_M{\asymp }_B\log (M+1)$, with ${\mu }_r=\lceil 3r/2\rceil$ for $r\ge 2$. A parity-compatible central window handles $r<L_M$, while a shrinking balanced window handles $r\ge L_M{\asymp }_B\log (M+1)$, no parity is used in this regime. The leading high-degree loss--gain factor is $\ee\,2^{-B}$; careful uniform bounds close the bootstrap for every $B>1/\log 2$. The formula for ${\mu }_r$ is explained below.

31 pages