汉明切片能量估计及其应用

中文摘要 AI 辅助

设 $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$，其中 $n\geq 3$，并设 $S_w\subseteq R$ 为其规范 $n$ 位二进制展开具有汉明重量 $w$ 的剩余类。我们特别获得了 $S_w$ 的加性能量的渐近公式 \\[ E(S_w)=\frac{\left|S_w\right|^4}{|R|}+ \mathcal{O}\left(|R|^3 n^{-3} \right), \\] 该公式对 $w$ 一致成立。误差项在阶上是最优的，对于 $w=\lfloor n/2+\sqrt{n} \rfloor$ 有匹配的下界。由此可知，当 $\prod_{j=1}^{3} \left|S_{w_j} \right| /\left(|R| n^{-3/5}\right)^3\to\infty$ 时，任意单位膨胀的三重和具有渐近一致的表示计数；当 $\left|S_{w_1}\right| \left|S_{w_2} \right|/\left(|R| n^{-3/4}\right)^2\to\infty$ 时，二重和具有渐近完全支撑。在证明中，模碰撞使用循环二进制进位自动机表示；这似乎是该问题领域中的一种新方法。

英文摘要

Let $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$, where $n\geq 3$, 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.

发表机构

• Purdue University（普渡大学）

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