Bounds for generalized Sidon sets

Let $\Gamma$ be an abelian group and $g \geq h \geq 2$ be integers. A set $A \subset \Gamma$ is a $C_h[g]$-set if given any set $X \subset \Gamma$ with $|X| = k$, 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 $g \geq h \geq 2$, we prove that if $A \subset \{1,2, \dots ,n \}$ is a $C_h[g]$-set in $\mathbb{Z}$, then $|A| \leq (g-1)^{1/h} n^{1 - 1/h} + O(n^{1/2 - 1/2h})$. We show that for any integer $n \geq 1$, there is a $C_3 [3]$-set $A \subset \{1,2, \dots , n \}$ with $|A| \geq (4^{-2/3} + o(1)) n^{2/3}$. We also show that for any odd prime $p$, there is a $C_3[3]$-set $A \subset \mathbb{F}_p^3$ with $|A| \geq p^2 - p$, which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer $h \geq 3$, there is a $C_h[h! +1]$-set $A \subset \{1,2, \dots , n \}$ with $|A| \geq ( c_h +o(1))n^{1-1/h}$. A set $A$ is a \emph{weak $C_h[g]$-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 $C_h[g]$-sets in $\{1,2, \dots , n \}$ for any $g \geq h \geq 2$. Lastly we obtain upper bounds on infinite $C_h[g]$-sequences. We prove that for any infinite $C_h[g$]-sequence $A \subset \mathbb{N}$, we have $A(n) = O ( n^{1 - 1/h} ( \log n )^{ - 1/h} )$ for infinitely many $n$, where $A(n) = | A \cap \{1,2, \dots , n \}|$.