Optimal Diameters of High Multiplicity g-Golomb Rulers

Abstract

A set $\mathcal{G}$ of integers is called a $g$-Golomb ruler of length $n$ if the difference between any two distinct elements of $\mathcal{G}$ is repeated at most $g$ times. If $g=1$, these are also called $B_2$-sets, Sidon sets, and Babcock sets. We define $G(g,n)$ to represent the minimum diameter of a $g$-Golomb Ruler. In this paper, we prove that for all $b\ge 1$, if $g \ge \frac{7}{4}\left(b^{3/2} -b\right)+1,$ then $G(g,g+b)=g+2b-2$. Sharper bounds are given for $b\le 18$. The main technique is through an arithmetic property of the integers that are \emph{not} in a $g$-Golomb ruler, leading us to introduce LM rulers, a new class of rulers where every distance $d$ occurs as a difference at most $d-1$ times. We show that the minimum diameter of an $n$-element LM ruler $L(n)$ is $\sqrt{8/9} \cdot (n-1)^{3/2} \le L(n) \le \frac{7}{4}\left((n+1)^{3/2}-(n+1)\right).$