Energy Estimation of the Hamming Slice and its Applications

Let $R=\mathbb{Z}\mathrm{/}(2^n-1)\mathbb{Z}$, where $n\ge 3$, and let $S_w\subseteq R$ be the residues whose canonical $n$-digit binary expansion has Hamming weight $w$. We obtain, in particular, an asymptotic formula for the additive energy of $S_w$ $$E(S_w)=\frac{{\mid S_w\mid }^4}{\mathrm{\mid }R\mathrm{\mid }}+\mathcal{O}\left(\mathrm{\mid }R{\mathrm{\mid }}^3{n}^{-3}\right),$$ which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\lfloor n\mathrm{/}2+\sqrt{n}\rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when ${\prod }_{j=1}^3\mid S_{w_j}\mid \mathrm{/}{\left(\mathrm{\mid }R\mathrm{\mid }{n}^{-3\mathrm{/}5}\right)}^3\to \mathrm{\infty }$, and that double sums have asymptotically full support when $\mid S_{w_1}\mid \mid S_{w_2}\mid \mathrm{/}{\left(\mathrm{\mid }R\mathrm{\mid }{n}^{-3\mathrm{/}4}\right)}^2\to \mathrm{\infty }$. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.