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Statement

Notation (p. 204). $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ are strictly increasing sequences of positive integers, $A(n)=\lvert A\cap\{1,\ldots,n\}\rvert$, and $\Gamma(N)$ is the set of the subsets of $\{1,\ldots,N\}$.

Infinite sets (pp. 204--205). An infinite sequence $B$ is a difference intersector set if the equation

$$a_x-a_y=b_z\qquad(1)$$

is solvable for every infinite sequence $A$ of positive lower asymptotic density, that is, if $B$ meets the difference set of each such $A$. It is a sum intersector set if, for every such $A$, the equation

$$a_x+a_y=b_z\qquad(2)$$

is solvable. The paper adds: "This terminology is due, partly, to R. Tijdeman." (p. 204). Equation (2) places no distinctness condition on $x$ and $y$.

Finite sets (p. 205). For a finite $B$ inside $\{1,\ldots,N\}$ (printed "$B\subset\Gamma(N)$"), $B$ is again called a difference intersector set if, for $A\in\Gamma(N)$,

$$A(N)>\varepsilon N\qquad(3)$$

implies the solvability of (1) "if $N$ is large in terms of $\varepsilon$". For sum intersector sets (3) is replaced by $A([N/2])>\varepsilon N$, since two elements of $A$ above $[N/2]$ have a sum above $N$, out of reach of $B$.

Examples quoted from Sárközy (pp. 205--206). The squares $\{1^2,2^2,\ldots\}$ and the shifted primes $\{2-1,3-1,5-1,\ldots,p-1,\ldots\}$ are difference intersector sets, by Sárközy's quantitative Theorems 1 and 2 (the paper's references [3] and [5]): for large $N$ and $A\in\Gamma(N)$, the bound $A(N)>c_1N(\log_2N)^{2/3}/(\log N)^{1/3}$ gives a solution of $a_x-a_y=z^2$ with $z>0$, and $A(N)>c_2N(\log_3N)^3\log_4N/(\log_2N)^2$ gives a solution of $a_x-a_y=p-1$, where $\log_kx$ is the $k$-fold iterated logarithm and $c_1,c_2$ are positive absolute constants. These are results of the cited papers and are not proved in this one.

Source. P. Erdős and A. Sárközy, On differences and sums of integers, II, Bull. Soc. Math. Grèce (N.S.) 18 (1977), no. 2, 204--223: the notation and definitions on pp. 204--205, Theorems 1 and 2 on pp. 205--206. The edition read is identified on the source card.

Read depth. Claims checked: the definitions and the statements of Theorems 1 and 2 were read clause by clause on the printed pages. Nothing here is independently reviewed.

Bears on

• Problem 439: the problem asks whether every finite coloring of the integers has a monochromatic pair $x\ne y$ with $x+y$ a square. These definitions are the density notions the paper works with; it shows on p. 209 that the squares are not a sum intersector set (the p. 209 remark). The paper does not pose the coloring question.