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

Abstract

Let $\mathbb{N}$ be the set of natural numbers and ${\mathcal{S}}_r=\{1^r,2^r,3^r,\cdots \}$ 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 $$f_r(n)=\#\{(w,m^r)\in {\mathcal{W}}_r\times {\mathcal{S}}_r:n=w+m^r\}.$$ 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 $$\sum _{n\le N}{f}_2(n)-N\gg N^{3/4-o(1)},$$ which improves the previous bound $N^{1/2}$ due to Ding, Sun, Wang and Xia.