The board / additive combinatorics / #1193
Erdős Problem #1193
SOLVED LEANdatabase updated 2026-04-13
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(A\subset \mathbb{N}\) and let \(g(n)\) be a non-decreasing function of \(n\) which is always \(>0\). Is the lower density of \[\{ n : 1_A\ast 1_A(n)=g(n)\}\] always \(0\)? The answer is trivially no to both questions: indeed if \(A=\mathbb{N}\) (assuming \(0\in\mathbb{N}\)) then \(1_A\ast 1_A(n)=n+1\) for all \(n\). Presumably Erdős had some additional restrictions on either \(g\) or \(A\) in mind, but these are not recorded in [Er80].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1193.Prizenone
Topicsadditive combinatorics
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1193
Attempts
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