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Claim. Vitaly Bergelson and David Simmons, New examples of complete sets, with connections to a Diophantine theorem of Furstenberg, Acta Arith. 177 (2017), no. 2, 101--131; arXiv:1507.02208 (v1 of 8 July 2015, v2 of 24 September 2016). A set of positive integers is complete when every sufficiently large integer is a sum of distinct elements of it, and strongly complete when it stays complete after any finite subset is removed. Theorem 1.23 (Theorem 1.22 in the first arXiv version): let $S_1,S_2,S_3,S_4$ be finite, pairwise disjoint subsets of $\mathbb N\setminus\{1\}$ with $\gcd(S_4)=1$ and $\sum_{a\in S_i}1/(a-1)\ge1$ for $i=1,2,3$. Then the set of all powers $a^n$, with $a\in S_1\cup S_2\cup S_3\cup S_4$ and $n\ge0$, is strongly complete. The argument was not reconstructed in this corpus.

Covers. The second question of Problem 124, for every $k\ge1$, at every tuple $d_1<\cdots<d_r$ that contains four such disjoint subsets: yes. The powers with exponent below $k$ form a finite set, so strong completeness makes every sufficiently large integer a sum of distinct powers $a^n$ with $n\ge k$ and $a$ in the union, and such a sum is $\sum_ic_ia_i$ with $a_i\in P(d_i,k)$, one term per base. Not covered: tuples without such subsets, in particular every tuple whose reciprocal sum is at most $3$, and the first question, which the theorem also answers at these tuples but which is claimed in full on Alexeev's page.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Arithmetica 177 (2017), no. 2, 101--131. The site's commentary (page last edited 1 December 2025) does not mention the paper and labels the problem OPEN, so no reviewed evidence is listed; a post on the site's thread of 23 September 2026 cites the theorem, with Fan's (claim page), as settling the second question when the reciprocal sum is large enough.