No exact on average additive complements of squares
Yuchen Ding, Zihan Zhang
TL;DR
The paper investigates whether the squares admit an exact on average additive complement by examining the representation-counting function $f(n)$ for sums $n=w+m^2$ with $w$ in a complement to squares. It proves a positive density result: there exists $c_W>0$ such that $\sum_{n\le N} f(n)-N \ge c_W N$ for infinitely many $N$, advancing Cilleruelo 1993 conjecture and enabling negative answers to questions of Ruzsa (2001) and Green (2017). The method hinges on a case analysis of the growth of $W(N)$ relative to $\sqrt{N}$, plus counting arguments and partial summation to obtain lower bounds, with a key contradiction argument in one case. The results are extended to $r$-th powers, showing analogous nonexistence of exact on average additive complements for $\{m^r\}$ and yielding related corollaries.
Abstract
Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}=\big\{1^2, 2^2, 3^2,\cdots\big\}$ the set of squares. Let $\mathcal{W}$ be an additive complement of $\mathcal{S}$ and $$ f(n)=\#\big\{(w,m^2)\in \mathcal{W}\times \mathcal{S}: n=w+m^2\big\}. $$ It is proved that there is a positive constant $c_{\mathcal{W}}$ (depending at most on $\mathcal{W}$) such that $$ \sum_{n\le N}f(n)-N\ge c_{\mathcal{W}}N $$ for infinitely many positive integers $N$, which makes some progress on a 1993 conjecture of Cilleruelo. As consequences of the this result, we answer negatively a 2001 problem of Ruzsa as well as a 2017 problem of Ben Green.