Sidon sets with $\Delta$-separated sumsets in additive number theory

Abstract

The nonempty set $A$ of integers is $\Delta$-separated if $|a-a^{\prime }|\ge \Delta$ for all $a,a^{\prime }\in A$ with $a\ne a^{\prime }$. The set $A$ is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. A $B_2$-set is also called a Sidon set. A $B_{h,\Delta }$-set is a $B_h$-set whose sumset $hA$ is $\Delta$-separated. Upper and lower bounds are obtained for the cardinality of the largest $B_{2,\Delta }$-sets contained in the integer interval $\{1,2,\dots ,n\}$, that is, sets $A\subseteq \{1,2,\dots ,n\}$ such that, if $a,b,c,d\in A$ and $\{a,b\}\ne \{c,d\}$, then $|(a+b)-(c+d)|\ge \Delta$.

Revised and corrected; 11 pages