Problem 253

Let $1\leq a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_{i+1}/a_i\to 1$ . If every infinite arithmetic progression contains infinitely many integers which are the sum of distinct $a_i$ then every sufficiently large integer is the sum of distinct $a_i$ .

Source: erdosproblems.com/253

Number theory

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An accepted solution exists. The statement is false.

DISPROVED (LEAN). Cassels's Theorem II ([Ca60], refereed) constructs a sequence with $(c_{n+1}-c_n)/c_n^{1/2+\varepsilon}\to0$ , infinitely many elements in every arithmetic progression, and sums of distinct elements of upper density at most $\varepsilon$ , so the implication fails; the site records the disproof as Cassels's, and the claim page (Cassels, 1959) carries the acceptance. The Lean suffix is the site's label for a 2026 formalization of Cassels's disproof in a public repository, registered by formal-conjectures and the community database, linked on the same claim page, neither built nor audited here.

Problem 253

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