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Claim. C. L. Stewart and R. Tijdeman, On infinite-difference sets, Canad. J. Math. 31 (1979), no. 5, 897–910, received 19 September 1977 and revised 3 August 1978; the issue is dated October 1979, the page name's date. For a strictly increasing sequence $A$ of non-negative integers the paper writes $D$ for the set of non-negative integers occurring infinitely often as a difference of two terms of $A$, the set $D(A)$ of Problem 332. Its Theorem 2 (p. 898) states that if $A$ has upper density $\varepsilon>0$, there are $r\le\varepsilon^{-\log3/\log2}$ integers $k_1,\dots,k_r$ with $\bigcup_j(D+k_j)\supseteq\mathbb N_0$; the paper deduces that $D$ has no gap longer than twice $\max_j|k_j|$, and shows by an example that $\max_j|k_j|$ cannot be bounded in terms of $\varepsilon$. The paper records that Prikry obtained the bounded-gaps result independently, citing a private communication. Ruzsa, On difference sets, Studia Sci. Math. Hungar. 13 (1978), 319–326, 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).

Covers. Every $A\subseteq\mathbb N$ of positive upper density, hence every $A$ of positive density, the condition the site's commentary credits. Not covered: sets of upper density zero; by the paper's Theorem 3, every set of non-negative integers containing $0$ is the infinite-difference set of some sequence of density zero.

Acceptance. Refereed: Canadian Journal of Mathematics 31 (1979). The site's commentary credits Prikry, Tijdeman, Stewart and others with the positive-density condition through the surveys [St78] and [Ti79], but the site labels the problem OPEN, so no reviewed evidence is listed.

Depends on. No page of this wiki.