A new upper bound on Ruzsa's number on the Erd\H os-Tur\'{a}n conjecture

TL;DR

This paper establishes a new upper bound of 192 for Ruzsa's number across all positive integers, improving previous bounds and contributing to the understanding of the Erdős-Turán conjecture.

Contribution

The paper provides the first universal upper bound of 192 for Ruzsa's number, refining earlier bounds and advancing the study of additive number theory.

Findings

Ruzsa's number R_m is bounded by 192 for all positive integers m.

The previous bound was R_m ≤ 288, now improved to 192.

This result narrows the possible values of Ruzsa's number, aiding in the analysis of the Erdős-Turán conjecture.

Abstract

In this note, we show that the Ruzsa number $R_m$ is bounded by $192$ for any positive integer $m$, which improved the prior bound $R_m\le 288$ given by Y.--G. Chen in 2008.