The Bose-Chowla argument for Sidon sets

Abstract

Let $h\ge 2$ and let $\mathcal{A}=(A_1,\dots ,A_h)$ be an $h$-tuple of sets of integers. For nonzero integers $c_1,\dots ,c_h$, consider the linear form $\phi =c_1{x}_1+c_2{x}_2+\cdots +c_h{x}_h$. The \emph{representation function} $R_{\mathcal{A},\phi }(n)$ counts the number of $h$-tuples $(a_1,\dots ,a_h)\in A_1\times \cdots \times A_h$ such that $\phi (a_1,\dots ,a_h)=n$. The $h$-tuple $\mathcal{A}$ is a \emph{$\phi$-Sidon system of multiplicity $g$} if $R_{\mathcal{A},\phi }(n)\le g$ for all $n\in \mathbf{Z}$. For every positive integer $g$, let $F_{\phi ,g}(n)$ denote the largest integer $q$ such that there exists a $\phi$-Sidon system $\mathcal{A}=(A_1,\dots ,A_h)$ of multiplicity $g$ with \[ A_i \subseteq [1,n] \qquad \text{and} \qquad |A_i| = q \] for all $i=1,\dots ,h$. It is proved that, for all linear forms $\phi$, \[ \limsup_{n\rightarrow \infty} \frac{F_{φ,g}(n)}{n^{1/h}} < \infty \] and, for linear forms $\phi$ whose coefficients $c_i$ satisfy a certain divisibility condition, \[ \liminf_{n\rightarrow\infty} \frac{F_{φ,h!}(n)}{n^{1/h}} \geq 1. \]

Minor changes; 10 pages

Cited by in corpus (1)

• Note on a problem of Nathanson related to the $\phi$-Sidon set