4 problems

Cilleruelo's conjecture. For any $r\ge 2$,

Let $\mathcal{W}$ be an additive complement of the set of squares, and let $f(n)$ denote the number of representations of $n$ as $w+m^2$ with $w\in\mathcal{W}$ and $m^2$ a square.…

Let $\mathcal{S}$ denote the set of squares, and let $\mathcal{W}=\{w_n\}_{n=1}^{\infty}$ be an additive complement of $\mathcal{S}$, with the elements of $\mathcal{W}$ indexed inc…

Sárközy–Szemerédi conjecture. There may exist exact additive complements such that