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Statement

Notation (p. 5-01). For a set $A$ of non-negative integers, $\mathcal D(A)$ is its ordinary-difference set, the non-negative integers that are differences of two elements of $A$, and $|E|_x$ is the number of elements of $E$ less than $x$.

Theorem 4 (p. 5-03), stated as a consequence of Theorem 6 of Stewart and Tijdeman's paper on infinite-difference sets. Let $\mathcal E$ be any countable set of infinite sets of positive integers, and let $\alpha$ be any number between $0$ and $1$. Then there is a set $A$ with density $\alpha$ such that

$$\limsup_{x\to\infty}\frac{|\mathcal D(A)\cap E|_x}{|E|_x}\le2\alpha \qquad\text{for every }E\in\mathcal E.$$

Consequence (p. 5-03). Taking $\mathcal E$ to be the set of all infinite arithmetic progressions and $\alpha$ any number between $0$ and $1/2$, there is a set of density $\alpha$ whose difference set contains no infinite arithmetic progression. The survey offers this against the expectation, which Theorems 1 and 2 might suggest, that $\mathcal D(A)$ contains an infinite arithmetic progression whenever $A$ has positive upper density.

Proof pointer

The survey gives no proof; it derives the theorem from Theorem 6 of Stewart and Tijdeman, On infinite-difference sets of sequences of positive integers (reference [14] of the survey, Canad. J. Math.). The consequence holds because an infinite arithmetic progression $E$ inside $\mathcal D(A)$ would give relative upper density $1>2\alpha$.

Read depth

Claims checked: Theorem 4 and its consequence were read clause by clause on the page image of the print. The proof is not in the survey and was not checked.

Dependencies

None in the corpus. External input: Theorem 6 of 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.