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Source. Lemma 2.2, p. 3, of Yuchen Ding, Yu-Chen Sun and Lilu Zhao, An improved upper bound on the Ruzsa number, arXiv:2607.06167 (2026), as identified on the source card.

Statement

$R_m$ is the Ruzsa number of $\mathbb Z/m\mathbb Z$ defined on p. 1 (see Theorem 1.1).

Lemma 2.2 (p. 3, quoted). "For $m\leqslant 132^2$, we have $R_m\leqslant 116$."

Here $m$ is a positive integer, as throughout the paper.

Proof pointer

Pages 2--4. The sequence $c_j=74j-\lfloor j^2/32\rfloor$ for $0\le j\le264$ (p. 2) has $c_{264}=17358>132^2-74$ (2.1) and consecutive gaps between $57$ and $74$ (2.2), and Lemma 2.1 (p. 3) records, by a finite computation the paper's appendix lists, that the largest number of pairs $(i,j)$ with $0\le i,j\le264$ and $c_i+c_j=n$, over integers $n$, is exactly $17$. For $m\le400$ the set $\{0,\ldots,19\}\cup\{20,40,\ldots,380\}$ modulo $m$ gives $R_m\le40$. For $400<m\le132^2$ the set is $\{0,\ldots,73\}\cup\{c_1,\ldots,c_\beta\}$ with $\beta$ the least index having $c_\beta\ge m-74$, and the count $74+2\cdot4+34=116$ bounds $\sigma_A(n)$ (pp. 3--4).

Dependencies

Lemma 2.1 of the same paper, a finite computation this page has not rerun. Read depth: claims checked; the statement was read clause by clause on p. 3 and the proof on pp. 3--4 for its structure only.

Bears on

• Problem 1192: background only, as one step of Theorem 1.1.