Problem 870

Let $k\geq 3$ and $A$ be an additive basis of order $k$ . Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$ ? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$ .)

Source: erdosproblems.com/870

Number theory
Additive bases

Wiki page

Status

ClaimedDisproved

A full solution has been claimed but not yet accepted. The statement is false.

OPEN, the site's label. The derived standing departs from it because a pending full claim, Turturean, answers the site's wording no for every $k\ge3$ , which makes the problem claimed as disproved. The claim is not accepted: its Lean development is not built or audited here, one outside validation of it is reported on the problem's thread, and there is no refereed publication.

ErNa88: Erdős, Paul and Nathanson, Melvyn B., Partitions of bases into disjoint unions of bases. J. Number Theory 29 (1988), 1-9; Problem 3 of Part 3 is this question.
ErNa79: Erdős, Paul and Nathanson, Melvyn B., Systems of distinct representatives and minimal bases in additive number theory. (1979), 89-107.
Ha56: Härtter, Erich, Ein Beitrag zur Theorie der Minimalbasen. J. Reine Angew. Math. (1956), 170-204.
Na74: Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324-333.

Problem 870

Status

Proof claims

Discussion