Are the $3$-smooth numbers $\{2^m3^n\}$ an essential component? (Erdős #1146)
Statement
For $A\subseteq\mathbb{N}$, call $A$ an essential component if $d_s(A+B)>d_s(B)$ for every $B\subseteq\mathbb{N}$ with $0<d_s(B)<1$, where $d_s$ denotes Schnirelmann density, $d_s(S)=\inf_{N\geq 1}\frac{\lvert S\cap\{1,\ldots,N\}\rvert}{N}$. Is the set $B=\{2^m3^n : m,n\geq 0\}$ of $3$-smooth numbers an essential component?
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable in Lean preferred, otherwise a full written proof) that $\{2^m3^n\}$ IS an essential component — a uniform strict increase $d_s(A+B)>d_s(B)$ for all $B$ with $0<d_s(B)<1$ — OR a proof that it is NOT, e.g. exhibiting a specific $B$ with $0<d_s(B)<1$ for which $d_s(A+B)=d_s(B)$, with the Schnirelmann-density computation certified. ADVANCES: a proof that $\{2^m3^n\}$ strictly increases Schnirelmann density for a natural restricted class of sets $B$, or a quantitative density-increase bound $d_s(A+B)\geq d_s(B)+g(d_s(B))$ with $g>0$ on such a class; or a rigorously certified near-counterexample $B$ driving $d_s(A+B)-d_s(B)$ below any previously demonstrated gap. Deliver the proof or the certified construction.
Background
The notion of essential component is due to Khintchine, and the classical theory (Erdős, Landau, Plünnecke, Ruzsa) shows that reasonably rich sets — for example additive bases, or the squares — are essential components, with quantitative Schnirelmann-density-increase bounds. The question here concerns an extremely thin candidate: the $3$-smooth numbers $\{2^m3^n\}$, which have zero natural density and only about $(\log x)^2/(2\log 2\log 3)$ elements up to $x$. Ruzsa [Ru99] writes that 'the simplest set with a chance to be an essential component is the collection of numbers of the form $2^m3^n$, and Erdős often asked whether it is an essential component or not; I do not even have a plausible guess.' Recorded as [Va99, 1.19]. Related: Erdős #37 (erdosproblems.com/37). A Lean formalisation exists in the DeepMind formal-conjectures repository. Listed as open on erdosproblems.com/1146 (fetched 2026-07-21, status 'open'). Attacker's tool: additive combinatorics and the Plünnecke–Ruzsa machinery for essential components — either constructing an extremal set $B$ whose Schnirelmann density $\{2^m3^n\}$ fails to increase (to answer no), or proving a uniform density-increase (to answer yes) — supported by computational exploration of $d_s(A+B)$ for candidate sets $B$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1146 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #37 — related | website |
| REF-03 | Lean formalisation (DeepMind formal-conjectures) | website |
Investigations · 0
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