For $A \subset N$ let $f_{r} (n)$ count the number of solutions to $n = a_{1} + \dots + a_{r}$ with $a_{i} \in A$ .Does there exist, for all $r \geq 2$ , a basis $A$ of order $r$ (so that $f_{r} (n) > 0$ for all large $n$ ) such that $n \leq x \sum f_{r} (n)^{2} ≪ x$ for all $x$ ?

Worked, still open.

additive combinatorics · open · possible · 0 attempts

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