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Claim. Corollary 1.12 of V. Bergelson and D. Simmons, New examples of complete sets, with connections to a Diophantine theorem of Furstenberg, Acta Arith. 177 (2017), no. 2, 101--131 (Corollary 1.11 in arXiv:1507.02208v1 of 8 July 2015, the date this page carries): for coprime integers $a,b\geq2$ and distinct integers $k_0=0,k_1,\dots,k_{4a-5}$, the set $\{a^nb^{k_m}: n\geq0,\ 0\leq m\leq4a-5\}$ is complete. Taking $k_m=m$ gives $K(a,b)\leq4a-5$, where $K(a,b)$ is the least $K$ for which $\{a^nb^l: n\geq0,\ 0\leq l\leq K\}$ is complete. Fang and Chen's quantitative form (p. 302) records the bound and remarks that the method seems to give no explicit threshold beyond which every integer is represented.

Covers. The set is a subset of $\{a^kb^l\}$, so the corollary proves the statement of Problem 246, in its corrected Statement, which takes $a,b\geq2$, in a stronger form with a linear bound on the exponent range.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Arithmetica, volume 177. The site's commentary on the problem does not cite the paper. The pending claim Song and Yue's bound presents itself as an improvement of this bound.