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arXiv · 2608.07416

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

Melvyn B. Nathanson
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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, $|x-x'| \geq \Delta$ for all $x,x' \in hA$ with $x\neq x'$. Upper and lower bounds are obtained for the cardinality of the largest $B_{2,\Delta}$-sets contained in $\{1,2,\ldots, n\}$, 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 $|(a+b)-(c+d)| \geq \Delta$.

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Melvyn B. Nathanson. 2026-08-07. Sidon sets with $\Delta$-separated sumsets in additive number theory. https://arxiv.org/abs/2608.07416

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