The Bose-Chowla argument for Sidon sets

TL;DR

This paper extends the Bose-Chowla argument to Sidon sets defined by linear forms, establishing bounds on the size of such sets within intervals, and demonstrating their asymptotic behavior.

Contribution

It generalizes the Bose-Chowla argument to a broader class of Sidon sets defined by linear forms, providing new bounds and asymptotic results.

Findings

Bounded the growth of maximum size of Sidon systems with fixed multiplicity.

Proved asymptotic lower bounds for the size of Sidon systems under certain divisibility conditions.

Established finiteness of the limsup of scaled maximum size for all linear forms.

Abstract

Let $h \geq 2$ and let ${ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of sets of integers. For nonzero integers $c_1,\ldots, c_h$, consider the linear form $\varphi = c_1 x_1 + c_2x_2 + \cdots + c_h x_h$. The \emph{representation function} $R_{ \mathcal{A},\varphi}(n)$ counts the number of $h$-tuples $(a_1,\ldots, a_h) \in A_1 \times \cdots \times A_h$ such that $\varphi(a_1,\ldots, a_h) = n$. The $h$-tuple $\mathcal{A}$ is a \emph{$\varphi$-Sidon system of multiplicity $g$} if $R_{\mathcal A,\varphi}(n) \leq g$ for all $n \in \mathbf{Z}$. For every positive integer $g$, let $F_{\varphi,g}(n)$ denote the largest integer $q$ such that there exists a $\varphi$-Sidon system $\mathcal {A} = (A_1,\ldots, A_h)$ of multiplicity $g$ with \[ A_i \subseteq [1,n] \qquad \text{and} \qquad |A_i| = q \] for all $i =1,\ldots, h$. It is proved that, for all linear forms $\varphi$, \[ \limsup_{n\rightarrow \infty} \frac{F_{\varphi,g}(n)}{n^{1/h}} < \infty \] and, for linear forms $\varphi$ whose coefficients $c_i$ satisfy a certain divisibility condition, \[ \liminf_{n\rightarrow\infty} \frac{F_{\varphi,h!}(n)}{n^{1/h}} \geq 1. \]