On the largest Sidon subset in a finite subset of $\mathbb{R}^N$

TL;DR

This paper establishes a new lower bound on the size of the largest Sidon subset within finite sets of integers and extends the result to subsets of Euclidean space, improving previous bounds and methods.

Contribution

It introduces a novel compression lemma and combines it with classical constructions to improve lower bounds on Sidon subsets in finite sets of integers and Euclidean spaces.

Findings

New lower bound: H(n) ≥ (1/3√3 + o(1))√n

Extension of bounds to subsets of ℝ^N using projection and approximation

Method applicable to other linear additive constraints like B_2[g] sets

Abstract

We obtain a new lower bound on the largest Sidon subset of an arbitrary finite set of integers. If $H(n)$ denotes the minimum, over all $n$-element subsets of $\mathbb Z$, of the largest Sidon subset they contain, we prove that $H(n) \geqslant \left(\frac{1}{3\sqrt 3}+o(1)\right)\sqrt n \gtrsim 0.19\sqrt n$. This improves a lower bound of Abbott related to a conjecture of Erd\H{o}s on Sidon subsets of arbitrary sets of integers. The main ingredient is a compression lemma which produces, from any finite set of integers, a large subset admitting an injective Freiman $2$-morphism into a cyclic group. Combined with Singer's covering of $\mathbb Z/(q^2+q+1)\mathbb Z$ by Sidon sets, this yields the stated bound. We further extend the result to finite subsets of $\mathbb R^N$, uniformly in the dimension, by means of a projection argument and a Dirichlet approximation preserving Sidon's equation. As a consequence, every set of $n$ points in $\mathbb R^N$ contains a Sidon subset of cardinality at least $\left(\frac{1}{3\sqrt 3}+o(1)\right)\sqrt n$. We also discuss an adaptation to $B_2[g]$ sets, obtaining a lower bound of order $\frac{1}{3\sqrt 3}\sqrt{gn}$, and explain how the method can be adapted to other linear additive constraints.