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Statement

Notation (p. 5-01). $\mathbb N_0$ is the set of non-negative integers; for $A\subseteq\mathbb N_0$, $\mathcal D_\infty(A)$ is the set of $d\in\mathbb N_0$ with $A\cap(A-d)$ infinite. Let $\mathfrak D$ be the collection of all sets $\mathcal D_\infty(A)$ with $A$ of positive upper density (p. 5-04; the print writes D).

Theorem 5 (p. 5-04), with a pointer to Stewart and Tijdeman's paper on infinite-difference sets. $\mathfrak D$ is a filter of the set of all subsets of $\mathbb N_0$.

Remarks (p. 5-04). $\mathfrak D$ is not an ultrafilter: there are disjoint sets with arbitrarily large gaps whose union is $\mathbb N_0$, and by Theorem 2 an infinite-difference set of a set of positive upper density has only bounded gaps. Since $\mathfrak D$ is a filter, the union and the intersection of two members of $\mathfrak D$ are again members, and if $A$ has positive upper density and $\mathcal D_\infty(A)\subseteq B\subseteq\mathbb N_0$, then $B=\mathcal D_\infty(C)$ for some $C$ of positive upper density. Neither ordinary-difference sets nor density-difference sets have this superset property: for the even non-negative integers $E$, $\mathcal D(E)=\mathcal D_0(E)=E$, while $E\cup\{1\}$ is not the ordinary-difference set of any set, and, with a pointer to Stewart and Tijdeman's paper on density-difference sets, no $A$ has $\mathcal D_0(A)=E\cup\{1\}$.

Proof pointer

The survey gives no proof; it points to Stewart and Tijdeman, On infinite-difference sets of sequences of positive integers (reference [14] of the survey, Canad. J. Math.).

Read depth

Claims checked: Theorem 5 and the remarks after it were read clause by clause on the page image of the print. The proof is not in the survey and was not checked.

Dependencies

Theorem 2 for the bounded-gaps remark. External input: the cited Stewart-Tijdeman paper.

Source. Cam L. Stewart, On difference sets of sets of integers, Séminaire Delange-Pisot-Poitou, Théorie des nombres, 19e année (1977/78), Fasc. 1, Exp. No. 5, 8 pp.; pages are cited by the print's own numbering 5-01 to 5-08, as on the source card.

Bears on

No Erdős problem page of the corpus cites this theorem.