Sidon sets in a union of intervals

TL;DR

This paper investigates the maximum size of Sidon sets within unions of integer intervals, providing bounds for unions of two and multiple intervals, advancing understanding of Sidon set distributions.

Contribution

It establishes new lower bounds for Sidon set sizes in unions of two intervals and extends bounds to unions of multiple intervals using the small differences technique.

Findings

Sidon set size at least 0.876√n in unions of two intervals

Bounds for Sidon sets in unions of k intervals

Application of small differences technique to union of intervals

Abstract

We study the maximum size of Sidon sets in unions of integers intervals. If $A\subseteq\mathbb{N}$ is the union of two intervals and if $\left| A \right|=n$ (where $\left| A \right|$ denotes the cardinality of $A$), we prove that $A$ contains a Sidon set of size at least $0, 876\sqrt{n}$. On the other hand, by using the small differences technique, we establish a bound of the maximum size of Sidon sets in the union of $k$ intervals.