The board / number theory / #29
Erdős Problem #29
PROVED LEANdatabase updated 2026-08-24
The statement as posed is true, and a proof exists. 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).
Is there an explicit construction of a set \(A\subseteq \mathbb{N}\) such that \(A+A=\mathbb{N}\) but \(1_A\ast 1_A(n)=o(n^\epsilon)\) for every \(\epsilon>0\)? The existence of such a set was asked by Sidon to Erdős in 1932. Erdős (eventually) proved the existence of such a set using probabilistic methods. This problem asks for a constructive solution. An explicit construction was given by Jain, Pham, Sawhney, and Zakharov [JPSZ24]. The formal statement records the existence of such a set; the linked formal proof exhibits an explicit construction.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/29.Prize$100
Topicsnumber theory, additive basis
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/29
Attempts
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