Erdős Problem #345

Let \(A\subseteq \mathbb{N}\) be a complete sequence, and define the threshold of completeness \(T(A)\) to be the least integer \(m\) such that all \(n\geq m\) are in \[P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}\] (the existence of \(T(A)\) is guaranteed by completeness). Is it true that there are infinitely many \(k\) such that \(T(n^k)>T(n^{k+1})\)? Here \(T(n^k)\) means the threshold of the positive \(k\)-th powers. Least-threshold witnesses ensure that the thresholds exist. We restrict to \(k\geq 1\), since the zeroth powers form \(\{1\}\), which is incomplete. Positive powers are complete [Ki17].