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arXiv · 2606.28651

On the Thickness of Infinite Generalized Sidon Sets, I

Kevin O'Bryant
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Abstract

Let $g \ge1$. A set $\mathcal{A}$ of nonnegative integers is a Sidon set if for each $d>0$ there is at most one pair $(a,b) \in \mathcal{A} \times \mathcal{A}$ with $d=a-b$. If there are at most $g$ pairs, then $\mathcal{A}$ is a $g$-Golomb ruler. We prove that if $\mathcal{A}$ is a $g$-Golomb ruler, then \[\liminf_{n\to\infty} \frac{ | \mathcal{A}\cap[0,n) | }{\sqrt{n/\log n}} \le \frac{2\sqrt g }{\sqrt{\log 2}},\] generalizing and sharpening results of Erd\H{o}s and Cilleruelo. There is a $g$-Golomb ruler $\mathcal{G}$ with \[\frac{\sqrt g }{\sqrt2} \le \limsup_{n\to\infty} \frac{ | \mathcal{G}\cap[0,n) | }{\sqrt n} \le \sqrt{g } ,\] generalizing a result of Kr\"uckeberg.

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Kevin O'Bryant. 2026-06-26. On the Thickness of Infinite Generalized Sidon Sets, I. https://arxiv.org/abs/2606.28651

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