Infinite Sidon-type sets for zero-sum linear forms

Abstract

Let $h\ge 2$, and let $\mathbf{b}=(b_1,\dots ,b_h)\in {\mathbb{Z}}^h$ be a zero-sum vector with nonzero coordinates. For a set $A=\{a_1<a_2<\cdots \}\subseteq \mathbb{N}$, let $r_{A,\mathbf{b}}(n)$ denote the number of $h$-tuples $(x_1,\dots ,x_h)$ of pairwise distinct elements of $A$ satisfying $b_1{x}_1+\cdots +b_h{x}_h=n$. We study density restrictions on sets $A$ for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case $\mathbf{b}=(c_1,-c_1,\dots ,c_k,-c_k)$, we prove that if $A(x)/(x/\log x{)}^{1/2k}\to \infty$, then $\frac{1}{x}{\sum }_{|n|\le x}{r}_{A,\mathbf{b}}(n)\to \infty$, whereas if $A(x)\gg x^{1/2k}$, then $\frac{1}{x}{\sum }_{|n|\le x}{r}_{A,\mathbf{b}}(n)\gg \log x$. This recovers Chen's theorem on $B_{2k}$-sequences. For general zero-sum vectors $\mathbf{b}$, we prove analogous bounds under gap conditions: if $a_{n+1}-a_n=o(n^{h-1}\log n)$, then $\frac{1}{x}{\sum }_{|n|\le x}{r}_{A,\mathbf{b}}(n)\to \infty$, whereas if $a_{n+1}-a_n\ll n^{h-1}$, then $\frac{1}{x}{\sum }_{|n|\le x}{r}_{A,\mathbf{b}}(n)\gg \log x$.

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