Bounds for generalized Sidon sets

Abstract

Let $\Gamma$ be an abelian group and $g\ge h\ge 2$ be integers. A set $A\subset \Gamma$ is a $C_h\left[g\right]$-set if given any set $X\subset \Gamma$ with $\left|X\right|=h$, and any set $\{k_1,\dots ,k_g\}\subset \Gamma$, at least one of the translates $X+k_i$ is not contained in $A$. For any $g\ge h\ge 2$, we prove that if $A\subset \{1,2,\dots ,n\}$ is a $C_h\left[g\right]$-set in $\mathbb{Z}$, then $\left|A\right|\le {(g-1)}^{1/h}{n}^{1-1/h}+O\left(n^{1/2-1/2h}\right)$. We show that for any integer $n\ge 1$, there is a $C_3\left[3\right]$-set $A\subset \{1,2,\dots ,n\}$ with $\left|A\right|\ge (4^{-2/3}+o\left(1\right))n^{2/3}$. We also show that for any odd prime $p$, there is a $C_3\left[3\right]$-set $A\subset {\mathbb{F}}_p^3$ with $\left|A\right|\ge p^2-p$, which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer $h\ge 3$, there is a $C_h[h!+1]$-set $A\subset \{1,2,\dots ,n\}$ with $\left|A\right|\ge (c_h+o\left(1\right))n^{1-1/h}$. A set $A$ is a weak $C_h\left[g\right]$-set if we add the condition that the translates $X+k_1,\dots ,X+k_g$ are all pairwise disjoint. We use the probabilistic method to construct weak $C_h\left[g\right]$-sets in $\{1,2,\dots ,n\}$ for any $g\ge h\ge 2$. Lastly we obtain upper bounds on infinite $C_h\left[g\right]$-sequences. We prove that for any infinite $C_h\left[g\right]$-sequence $A\subset \mathbb{N}$, we have $A\left(n\right)=O\left(n^{1-1/h}{(logn)}^{-1/h}\right)$ for infinitely many $n$, where $A\left(n\right)=|A\cap \{1,2,\dots ,n\}|$.