On the size of finite Sidon sets

TL;DR

This paper improves the lower bounds on the size of finite Sidon sets, providing tighter estimates for their diameter and maximum size within a range, using a simpler method than previous approaches.

Contribution

It introduces a simpler method to improve bounds on Sidon sets' diameter and size, refining previous numerical results with more straightforward reasoning.

Findings

Lower bound on Sidon set diameter: at least $k^2 - 1.99405 k^{3/2}$ for large $k$

Maximum size of Sidon subsets in $oxed{1,2,...,n}$ is at most $n^{1/2} + 0.99703 n^{1/4}$ for large $n$

Methodologically simpler approach than prior work by Balogh-F"uredi-Roy

Abstract

A Sidon set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d$. We improve on the lower bound on the diameter of a Sidon set with $k$ elements: if $k$ is sufficiently large and ${\cal A}$ is a Sidon set with $k$ elements, then $diam({\cal A})\ge k^2-1.99405 k^{3/2}$. Alternatively, if $n$ is sufficiently large, then the largest subset of $\{1,2,\dots,n\}$ that is a Sidon set has cardinality at most $n^{1/2}+0.99703 n^{1/4}$. While these are only slight numerical improvements on Balogh-F\"uredi-Roy (arXiv:2103:15850v2), we use a method that is logically simpler.