An improved upper bound on the Ruzsa number

Abstract

Let $R_m$ be the least positive integer $r$ such that there exists a set $A\subseteq \mathbb{Z}_{m}$ with $A+A=\mathbb{Z}_m$ for which the number of ordered solutions of $n=x+y$ with $x,y\in A$ is at most $r$ for every $n\in \mathbb{Z}_m$. In this note we prove that $R_m\leqslant 128$ for every positive integer $m$, improving the previous bound $R_m\leqslant 192$.

Disclosure

“0 30 7 10 14 4 0 28 8 12 12 4 0 28 Acknowledgments The author acknowledges the use of OpenAI’s ChatGPT during the preparation of this manuscript. This work is supported by the National Key Research and Development Program of China (Grant No. 2021YFA1000700) and the National Natural Science Foundation of China (Grant No. 12471088).”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-24T00:34:09.244Z

Structural counts