Erdős Problem #1146
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
We say that \(A\subset \mathbb{N}\) is an essential component if \(d_s(A \oplus B)>d_s(B)\) for every \(B\subset \mathbb{N}\) with \(0<d_s(B)<1\) where \(d_s\) is the Schnirelmann density. Here, the sumset is the appropriate one for Schnirelmann density, \(A \oplus B = \{a+b \mid a \in A \cup \{0\}, b \in B \cup \{0\}\}\) (i.e. \((A \cup \{0\}) + (B \cup \{0\})\)). This avoids the trivial case where the sumset misses \(1\) simply because neither \(A\) nor \(B\) contains \(0\).
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#1146 remains open; a counterexample must have vanishing simultaneous \(2^m3^n\)-boundary along density-minimizing prefixes, all eventually periodic competitors are excluded, and exact optimization gi
PARTIAL