k-fold Sidon sets

Abstract

Let $k\ge 1$ be an integer. A set $A\subset \mathbb{Z}$ is a $k$-fold Sidon set if $A$ has only trivial solutions to each equation of the form $c_1{x}_1+c_2{x}_2+c_3{x}_3+c_4{x}_4=0$ where $0\le |c_i|\le k$, and $c_1+c_2+c_3+c_4=0$. We prove that for any integer $k\ge 1$, a $k$-fold Sidon set $A\subset [N]$ has at most $(N/k{)}^{1/2}+O((Nk{)}^{1/4})$ elements. Indeed we prove that given any $k$ positive integers $c_1<\cdots <c_k$, any set $A\subset [N]$ that contains only trivial solutions to $c_i(x_1-x_2)=c_j(x_3-x_4)$ for each $1\le i\le j\le k$, has at most $(N/k{)}^{1/2}+O((c_k^{2}N/k{)}^{1/4})$ elements. On the other hand, for any $k\ge 2$ we can exhibit $k$ positive integers $c_1,\dots ,c_k$ and a set $A\subset [N]$ with $|A|\ge (\frac{1}{k}+o(1))N^{1/2}$, such that $A$ has only trivial solutions to $c_i(x_1-x_2)=c_j(x_3-x_4)$ for each $1\le i\le j\le k$.

arXiv comment8 pages