On Sidon sets in a random set of vectors

TL;DR

This paper investigates the size and count of Sidon sets within multi-dimensional integer grids, providing asymptotic estimates for their maximum size and total number, especially in random subsets.

Contribution

It offers the first asymptotic estimates for the number of Sidon sets and their maximum size in random subsets of multi-dimensional grids.

Findings

Logarithm of the number of Sidon sets is Theta(n^{d/2})

Maximum size of Sidon sets in random subsets is estimated

Constants depend only on the dimension d

Abstract

For positive integers $d$ and $n$, let $[n]^d$ be the set of all vectors $(a_1,a_2,\dots, a_d)$, where $a_i$ is an integer with $0\leq a_i\leq n-1$. A subset $S$ of $[n]^d$ is called a \emph{Sidon set} if all sums of two (not necessarily distinct) vectors in $S$ are distinct. In this paper, we estimate two numbers related to the maximum size of Sidon sets in $[n]^d$. First, let $\mathcal{Z}_{n,d}$ be the number of all Sidon sets in $[n]^d$. We show that $\log (\mathcal{Z}_{n,d})=\Theta(n^{d/2})$, where the constants of $\Theta$ depend only on $d$. Next, we estimate the maximum size of Sidon sets contained in a random set $[n]^d_p$, where $[n]^d_p$ denotes a random set obtained from $[n]^d$ by choosing each element independently with probability $p$.