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Statement

Notation (p. 5-01). $\mathbb N_0$ is the set of non-negative integers; for $A\subseteq\mathbb N_0$ and $d\in\mathbb N_0$, $A[d]=A\cap(A-d)$, $\mathcal D_0(A)$ is the set of $d$ with $\overline d(A[d])>0$, and $\overline d$, $\underline d$ are upper and lower density (the print writes $d^-$ and $d_-$).

Theorem 6 (p. 5-04), with a pointer to Stewart and Tijdeman's paper on density-difference sets. If $A$ and $B$ are subsets of $\mathbb N_0$, then there is a set $C\subseteq\mathbb N_0$ such that $\mathcal D_0(C)=\mathcal D_0(A)\cap\mathcal D_0(B)$ and $\overline d(C[d])\ge\overline d(A[d])\cdot\overline d(B[d])$ for every $d\in\mathbb N_0$.

Consequence (p. 5-04). Taking $d=0$ and applying display (2) of Theorem 2 to $C$ gives

$$\underline d\bigl(\mathcal D_0(A)\cap\mathcal D_0(B)\bigr)\ge \bigl[(\overline d(A)\,\overline d(B))^{-1}\bigr]^{-1}.$$

The survey states that this inequality is best possible, with a pointer to Stewart and Tijdeman's paper on infinite-difference sets, and that Ruzsa also proved it. The survey also records (p. 5-04) that the union and the intersection of two density-difference sets are again density-difference sets.

Proof pointer

The survey gives no proof; it points to Stewart and Tijdeman, On density-difference sets of sequences of integers (reference [15] of the survey, then to appear).

Read depth

Claims checked: Theorem 6 and the displayed 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

Theorem 2, through display (2), for the consequence. 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.