Cross representations of additive complements of $r$-th powers

TL;DR

This paper investigates the structure of additive complements of $r$-th powers, proving new lower bounds on their representation functions that generalize previous results for squares.

Contribution

It extends known bounds for the sum of representation functions of additive complements from squares to general $r$-th powers, confirming a conjecture for all $r \\ge 2$.

Findings

Established lower bounds for the sum of representation functions for all $r \\ge 2$

Improved the bound for the case $r=2$ by including a logarithmic factor

Confirmed a 1993 conjecture of Cilleruelo for general $r$

Abstract

Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}_r=\big\{1^r, 2^r, 3^r,\cdots\big\}$ the set of $r$-th powers, where $r\ge 2$ is a natural number. Let $\mathcal{W}_r$ be an additive complement of $\mathcal{S}_r$ and $$ f_r(n)=\#\big\{(w,m^r)\in \mathcal{W}\times \mathcal{S}_r: n=w+m^r\big\}. $$ Motivated by a 1993 conjecture of Cilleruelo, we show that $$ \sum_{n\le N}f_r(n)-N\gg_r N^{1-\frac{1}{r}}. $$ Previously, the bound was only proved for $r=2$. In the case $r=2$, the lower bound above can be made more explicit as $$ \sum_{n\le N}f_2(n)-N\gg N^{1/2}(\log N)^{\delta} $$ for some absolute constant $\delta>0$, which improves a $\log$ factor upon a recent result of Ding, Sun, Wang and Xia.