An optimal lower bound for the size of the restricted sumsets containing powers

Abstract

Let $\epsilon >0$ be a fixed real number and $r\ge 2$ be an integer. In 2023, Yu, Chen and Chen proved that for any sufficiently large positive integer n, if $A\subseteq [1,n]$ with $\gcd A=1$ and $\left|A\right|>(1/m(r)+\epsilon )n$, then there is a power of r that can be represented as the sum of distinct elements of A, where $m\left(r\right)$ is a computable positive integer only related to r. In this paper, we improve this result for $r\ge 3$. We prove that the condition $\left|A\right|>(1/m(r)+\epsilon )n$ can be replaced by $\left|A\right|>n/m\left(r\right)+f\left(r\right)$, where $f\left(r\right)$ is a computable positive integer only related to r. We will also show that this lower bound is optimal, namely, for infinitely many positive integers n, there exists $B\subseteq [1,n]$ with $\gcd B=1$ and $\left|B\right|=n/m\left(r\right)+f\left(r\right)$ such that no power of r can be represented as the sum of distinct elements of B. This also generalizes a result in which $r=2$ obtained by Yang and Zhao.