Problem 332

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claims/: The 2 claim pages of Problem 332, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded gaps?

Status. Open. The site credits Prikry, Tijdeman, Stewart and others with the sufficient condition that $A$ has positive density. Theorem 2 of Stewart and Tijdeman (Canad. J. Math. 1979, refereed) proves that positive upper density suffices, the accepted partial claim on its claim page; Prikry's independent proof, which they cite as a private communication, was not published and has no page. A note linked from the thread on 4 May 2026, posted as an observation by GPT 5.5 pro, claims positive upper Banach density, a pending partial claim on its claim page.

Source. erdosproblems.com/332, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #332, https://www.erdosproblems.com/332.

References.

• [St78] Stewart, Cam L., On difference sets of sets of integers. Séminaire Delange-Pisot-Poitou, 19e année: 1977/78, Théorie des nombres, Fasc. 1 (1978), Exp. No. 5, 8.
• [Ti79] Tijdeman, R., Distance sets of sequences of integers. Proceedings, Bicentennial Congress Wiskundig Genootschap (Vrije Univ., Amsterdam, 1978), Part II (1979), 405-415.

Formalization. Statement in formal-conjectures.

Current assessment

The question, as the site states it (page last edited 28 October 2025): which conditions on $A\subseteq\mathbb N$ ensure that $D(A)$, the set of differences occurring infinitely often in $A$, has bounded gaps? The site's commentary credits Prikry, Tijdeman, Stewart and others, through the surveys [St78] and [Ti79], with the sufficient condition that $A$ has positive density, and asks further which conditions give $D(A)$ positive density, a divergent sum of reciprocals, or just $D(A)\neq\emptyset$. Theorem 2 of Stewart and Tijdeman (Canad. J. Math. 31 (1979), refereed) proves that if $A$ has upper density $\varepsilon>0$, then at most $\varepsilon^{-\log3/\log2}$ translates of $D(A)$ cover the non-negative integers, so $D(A)$ has bounded gaps; this is the accepted partial claim on its claim page. The paper cites Prikry's independent proof as a private communication; it was not published and has no page. Ruzsa refined the covering to at most $1/\varepsilon$ translates of the set of $d$ for which $A\cap(A+d)$ has positive upper density, as Theorem 2 of the survey [St78] records (library card). A three-page note linked from the thread on 4 May 2026, posted by the forum user aditya as an observation by GPT 5.5 pro, claims that positive upper Banach density suffices, a weakening of the positive-density condition; it is unrefereed and names no author, the pending partial claim on its claim page. Belgikar, Bergelson, Black and Kruzel (arXiv:2412.01185) reprove and generalize the Stewart-Tijdeman and Ruzsa theorems by pointwise ergodic theory, with versions for amenable groups (library card); the pending claim page cites their Theorem 4.1 as a second route to the Banach-density statement. Both claims give sufficient conditions, the form of answer the problem asks for, and neither claims a characterization, so the problem stays open with one accepted partial claim and one pending partial claim.

Search scope. As of 2026-10-07 the site's page lists no proof claim, its discussion thread holds one comment, of 4 May 2026, the posting that the pending claim page records, and the formal-conjectures statement file states the question as a single open theorem whose sufficient condition is left to be supplied, with no formal proof.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• belgikar_2024_new_applications_ergodic_theory_sets_differences
• stewart_1978_difference_sets_sets_integers
• stewart_1978_difference_sets_sets_integers / theorem_2