The number of rational numbers determined by large sets of integers

TL;DR

This paper establishes lower bounds on the number of rational numbers generated by large integer subsets and explores their limitations, with applications to sequences and differences.

Contribution

It provides a new lower bound for the count of rational numbers from large integer subsets and constructs examples showing the bound's optimality.

Findings

Lower bound of $( ext{size of A} imes ext{size of B})^{1+ ext{epsilon}}$ for rational numbers

Construction of examples demonstrating the bound's sharpness

Application to differences in product sequences of integers

Abstract

When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, 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 S\'ark\"ozy concerning the differences of consecutive terms of the product sequence of a given integer sequence.