Erdős Problem #1192 — Rényi and I proved by the probabilistic method that there is a sequence $a_k < ck^r$ for which … Probably (10) holds even for a basis of order $r$.

Rényi and I proved by the probabilistic method that there is a sequence $a_k < ck^r$ for which

$$\sum_{n=1}^{x} f_r(n)^2 < Cx.$$

Probably (10) holds even for a basis of order $r$. In other words there is a sequence $A$ satisfying (10) and $f_r(n) > 0$ for every $n$.