Energy Estimation of the Hamming Slice and its Applications

Abstract

Let $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$, 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 $S_w$ \[ E(S_w)=\frac{\left|S_w\right|^4}{|R|}+ \mathcal{O}\left(|R|^3 n^{-3} \right), \] which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\lfloor n/2+\sqrt{n}\rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when ${\prod }_{j=1}^3\left|S_{w_j}\right|/{\left(|R|n^{-3/5}\right)}^3\to \infty$, and that double sums have asymptotically full support when $\left|S_{w_1}\right|\left|S_{w_2}\right|/{\left(|R|n^{-3/4}\right)}^2\to \infty$. 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.