On the Thickness of Infinite Sidon Sets

Let $\gamma \ge 1$. A set $A$ of nonnegative integers is a Sidon set if for each $d>0$ there is at most one pair $(a,b)\in A\times A$ with $d=a-b$. If there are at most $\gamma$ pairs, then $A$ is a $\gamma$-Golomb ruler. We prove that if $A$ is a $\gamma$-Golomb ruler, then $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\mathrm{\mid }A\cap [0,n)\mathrm{\mid }}{\sqrt{n\mathrm{/}\log n}}\le \frac{2\sqrt{\gamma }}{\sqrt{\log 2}},$$ generalizing and sharpening results of Erdős and Cilleruelo. There is a $\gamma$-Golomb ruler $G$ with $$\frac{\sqrt{\gamma }}{\sqrt{2}}\le \underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\mathrm{\mid }G\cap [0,n)\mathrm{\mid }}{\sqrt{n}}\le \sqrt{\gamma },$$ generalizing a result of Krückeberg.