An upper bound for weak $B_k$-sets

TL;DR

This paper establishes an upper bound on the size of weak B_k-sets, which are subsets of integers avoiding solutions to specific additive equations, showing their maximum possible size relative to N and k.

Contribution

It provides a new upper bound for the size of weak B_k-sets, advancing understanding of their structure and limitations in additive combinatorics.

Findings

Upper bound of |A| ≪ k^{3/2} N^{1/k} for weak B_k-sets

Demonstrates limitations on the size of sets avoiding certain additive solutions

Advances theoretical understanding of additive combinatorics constraints

Abstract

We prove that if $A\subseteq \{ 1,2,\dots, N \}$ 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 $|A|\ll {k^{3/2}}N^{1/k}.$