Problem 1146

We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density.

Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component?

Source: erdosproblems.com/1146

Number theory

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Open

No claim settles this problem.

Ru99: Ruzsa, I., Erdős and the Integers. Journal of Number Theory 79 (1999), 115--163, doi:10.1006/jnth.1999.2395; § 12, Random sets: the definition of an essential component, printed p. 146 (PDF p. 32 of the publisher's open-archive PDF), and the question whether the numbers $2^m3^n$ form one, attributed to Erdős's repeated asking and left unanswered, its author having no plausible guess, printed p. 147 (PDF p. 33); the survey records no result on the set itself. Library home: Ruzsa 1999, with the passage paged on Ruzsa 1999, Question (p. 147).
Va99: Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999); the site's source key for this problem is [Va99, 1.19].

Problem 1146

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