Problem 1146

../

---

Statement. 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?

Formulation. The sum $A+B$ is read as in Schnirelmann's theory, with $0$ adjoined to each set: $A\oplus B=\{a+b:a\in A\cup\{0\},\ b\in B\cup\{0\}\}$. That is the setting of Ruzsa's survey, the site's source. Its inequality of Erdős, from which it deduces that every basis is an essential component, is stated for a basis containing $0$. A reply in the site's thread (31 May 2026) gives the same reading. The formal-conjectures statement has used it since its correction of 10 June 2026. So read, the question is open.

With the ordinary sumset the wording has a trivial negative answer. When $0\notin C$, every element of $\{2^m3^n\}+C$ is at least $2$, so $d_s(\{2^m3^n\}+C)=0$. A test set such as $C=\{1\}\cup\{2,4,6,\ldots\}$, with $d_s(C)=1/2$, therefore violates the definition; a thread post of 31 May 2026 makes this observation.

Status. Open.

Source. erdosproblems.com/1146, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1146, https://www.erdosproblems.com/1146.

References.

• [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_erdos_integers, with the passage paged on question_p147.
• [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].

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• ruzsa_1999_erdos_integers
• ruzsa_1999_erdos_integers / question_p147
• gyory_2020_additive_multiplicative_decompositions_sets_integers_restricted
• gyory_2020_additive_multiplicative_decompositions_sets_integers_restricted / theorem_1_1