Existence of a Sidon set for the distinct distance constant

TL;DR

This paper demonstrates the existence of special Sidon sets that maximize certain functions, including the reciprocal sum equaling the distinct distance constant, and improves bounds for this constant.

Contribution

It proves the existence of Sidon sets that achieve the maximum reciprocal sum and enhances the known bounds for the distinct distance constant.

Findings

Existence of Sidon sets with maximal reciprocal sum.

Improved bounds for the distinct distance constant.

Applications of Sidon sets and $B_2[g]$-sets highlighted.

Abstract

We highlight a certain compactness of Sidon sets and $B_2[g]$-sets and provide several applications. Notably, we prove the existence of such sets that maximize certain functions. In particular, we show the existence of a Sidon set whose reciprocal sum is equal to the distinct distance constant. We also improve the best known bounds for this constant.