Adjacent comparison bounds and extremal sets for Ruzsa numbers

Abstract

Let $m$ be a positive integer and ${\mathbb{Z}}_m$ the residue class ring modulo $m$. The Ruzsa number $R_m$ is defined to be the least integer $r$ such that there is a subset $\mathcal{A}$ of ${\mathbb{Z}}_m$ satisfying $1\le \ddot{I}{ƒ}_{\mathcal{A}}(n)\le r$ for any $n\in {\mathbb{Z}}_m$, where $$\ddot{I}{ƒ}_{\mathcal{A}}(n)=\#\{(a,a^{\prime })\in {\mathcal{A}}^2:a+a^{\prime }\equiv n(\mathrm{mod}m)\}.$$ Motivated by a 2024 conjecture of Ding and Zhao, we prove $|R_{m+1}-R_m|\le 144.$ Let $\mathcal{A}$ be a subset of ${\mathbb{Z}}_m$ satisfying $1\le \ddot{I}{ƒ}_{\mathcal{A}}(n)\le R_m$ for any $n\in {\mathbb{Z}}_m$. We also give nontrivial bounds for the size of $\mathcal{A}$. Additionally, we provide exact values of $R_m$ for all $m\le 100$, which substantially extends the table of values given by Sándor and Yang in 2017. Finally, we pose several related problems and prove some partial results.