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

Abstract

The nonempty set $A$ of integers is $\Delta$-separated if $|a-a'| \geq \Delta$ for all $a,a' \in A$ with $a \neq a'$. 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 $\{1,2,\ldots, n\}$, that is, sets $A \subseteq \{1,2,\ldots, n\}$ such that, if $a,b,c,d \in A$ and $\{a,b\} \neq \{c,d\}$, then $|(a+b)-(c+d)| \geq \Delta$.