On the Thickness of Infinite Generalized Sidon Sets, I

Abstract

Let $g\ge 1$. 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 Erdős 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 Krückeberg.

10 pages; v3 has new title and refactored proof