An Upper Bound for Weak $B_k$-Sets

Abstract

We prove that if $A\subseteq [N]$ does not contain any solution to the equation $x_1+\dots+x_k=y_1+\dots+y_k$ with distinct $x_1,\dots,x_k,y_1,\dots,y_k\in A$, then $|A|\le 16 {k^{3/2}}N^{1/k},$ provided $N\ge (2k^{2})^{2k}$. This problem was first considered by Ruzsa, and this upper bound improves the previously best known upper bound of $(\frac{1}{4} + o_k (1)) k^2 N^{1/k}$ which was proved by Timmons.