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

Let $q_m=2m\mathrm{/}(m+1)$ and put $${\beta }_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\dots ,$$ where $\log$ is the natural logarithm. We give a proof scheme showing that, for every $\epsilon >0$, there is $C_{\epsilon }<\mathrm{\infty }$ such that every complex-valued function $f:\{-1,1{\}}^n\to \text{C}$ of Fourier degree at most $m$ satisfies $${\left(\sum _{A\subseteq [n]}\text{abs}{\text{wh}f(A)}^{q_m}\right)}^{1\mathrm{/}{q}_m}\le C_{\epsilon }{m}^{{\beta }_0+\epsilon }\text{norm}{f}_{\mathrm{\infty }}\mathrm{.}$$ The improvement over the $m^9$ estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales $m$, $m^{3\mathrm{/}2}$, and $m^{5\mathrm{/}3}$. These losses are encoded in the weight $M^{{\mu }_r\mathrm{/}r}{r}^B\min \{r,L_M{\}}^8$, $L_M{\asymp }_B\log (M+1)$, with ${\mu }_r=\lceil 3r\mathrm{/}2\rceil$ 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 $\text{ee}{2}^{-B}$; careful uniform bounds close the bootstrap for every $B>1\mathrm{/}\log 2$. The formula for ${\mu }_r$ is explained below.