---

Statement

Notation (p. 5-01). $\mathbb N_0$ is the set of non-negative integers. For $A\subseteq\mathbb N_0$ and an integer $d$, $A[d]=A\cap(A-d)$. The ordinary-difference set is $\mathcal D(A)=\{d\in\mathbb N_0: A[d]\ne\emptyset\}$, the infinite-difference set is $\mathcal D_\infty(A)=\{d\in\mathbb N_0: |A[d]|=\infty\}$ and the density-difference set is $\mathcal D_0(A)=\{d\in\mathbb N_0: \overline d(A[d])>0\}$. With $|A|_x$ the number of elements of $A$ less than $x$, the upper density is $\overline d(A)=\limsup_{x\to\infty}|A|_x/x$ and the lower density is $\underline d(A)=\liminf_{x\to\infty}|A|_x/x$ (the print writes $d^-(A)$ and $d_-(A)$).

Theorem 2 (p. 5-02), credited to Ruzsa, refining work of Stewart and Tijdeman. Let $A$ have positive upper density $\varepsilon$. Then there are $r$ integers $k_1,\ldots,k_r$ with $r\le\varepsilon^{-1}$ such that $\bigcup_{j=1}^r(\mathcal D_0(A)+k_j)\supseteq\mathbb N_0$.

Sharpness (pp. 5-02 to 5-03). The bound on $r$ is best possible: for the multiples $A_\ell$ of $\ell$, $\mathcal D_0(A_\ell)=A_\ell$ and $\ell$ translates are needed. The size of the shifts is not bounded in terms of $\varepsilon$: the set $A_t$ of integers $3nt+i$ ($i=1,\ldots,t$, $n=0,1,2,\ldots$) has $d(A_t)=1/3$, while $\mathcal D_0(A_t)$ is the set of non-negative integers $3nt\pm i$ ($i=0,\ldots,t$), which has infinitely many gaps of length $t$, so $\max_j|k_j|\ge[t/2]$.

Display (2) (p. 5-03). The survey records as an immediate consequence of Theorem 2 that if $\overline d(A)=\varepsilon$ then $\underline d(\mathcal D_0(A))\ge[\varepsilon^{-1}]^{-1}$. Since $\mathcal D(A)\supseteq\mathcal D_\infty(A)\supseteq\mathcal D_0(A)$, each of the three difference sets of $A$ then has lower density at least the upper density of $A$.

Proof pointer

The survey gives no proof; it attributes the theorem to Ruzsa, On difference sets (reference [10] of the survey, then to appear), refining Stewart and Tijdeman, On infinite-difference sets of sequences of positive integers (reference [14], Canad. J. Math.). Display (2) follows, in the corpus's reading, because each translate $\mathcal D_0(A)+k_j$ has at most $|\mathcal D_0(A)|_x+|k_j|$ elements below $x$, so covering $\mathbb N_0$ by $r$ of them gives $r\,\underline d(\mathcal D_0(A))\ge1$, and $r$, an integer at most $\varepsilon^{-1}$, is at most $[\varepsilon^{-1}]$.

Read depth

Claims checked: the definitions, Theorem 2, the two examples and display (2) were read clause by clause on the page images of the print. The proof is not in the survey and was not checked.

Dependencies

None in the corpus. External input: Ruzsa's cited 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

• Problem 332: the problem's $D(A)$ is the survey's $\mathcal D_\infty(A)$, which contains $\mathcal D_0(A)$; so for $A$ of positive upper density the theorem gives finitely many translates of $D(A)$ covering $\mathbb N_0$, and the survey notes (p. 5-04) that by Theorem 2 this set has only bounded gaps. Display (2) gives it lower density at least $[\varepsilon^{-1}]^{-1}$. Both are sufficient conditions under positive upper density, not a characterization of the sets $A$ the problem asks about.