Adjacent comparison bounds and extremal sets for Ruzsa numbers

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 $1\le {\sigma }_{\mathcal{A}}(n)\le r$ for any $n\in {\mathbb{Z}}_m$, where $${\sigma }_{\mathcal{A}}(n)=\mathrm{\#}\{(a,a^{\mathrm{\prime }})\in {\mathcal{A}}^2:a+a^{\mathrm{\prime }}\equiv n(\mathrm{m}\mathrm{o}\mathrm{d}m)\}\mathrm{.}$$ Motivated by a 2024 conjecture of Ding and Zhao, we prove $\mathrm{\mid }{R}_{m+1}-R_m\mathrm{\mid }\le 144.$ Let $\mathcal{A}$ be a subset of ${\mathbb{Z}}_m$ satisfying $1\le {\sigma }_{\mathcal{A}}(n)\le R_m$ 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.