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

Abstract

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 $A$ whose sumset $hA$ is $\Delta$-separated, that is, $|x-x^{\prime }|\ge \Delta$ for all $x,x^{\prime }\in hA$ with $x\ne x^{\prime }$. Upper and lower bounds are obtained for the cardinality of the largest $B_{2,\Delta }$-sets contained in $\{1,2,\dots ,n\}$, 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 improved with new open problems; 11 pages