The number of rational numbers determined by large sets of integers

Abstract

When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A|\ge \alpha X$ and $|B|\ge \beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A\times B$ is $\gg (\alpha \beta {)}^{1+\epsilon }XY$ for any $\epsilon >0$, where the implied constant depends on $\epsilon$ alone. We then construct examples that show that this bound cannot in general be improved to $\gg \alpha \beta XY$. We also resolve the natural generalisation of our problem to arbitrary subsets $C$ of the integer points in $[1,X]\times [1,Y]$. Finally, we apply our results to answer a question of Sárközy concerning the differences of consecutive terms of the product sequence of a given integer sequence.

11 pages

Cited by in corpus (2)

• A note on product sets of rationals
• On the distribution of the digits of quotients of integers and primes