Ratio sets of random sets

Abstract

We study the typical behaviour of the size of the ratio set A / A for a random subset $A\subset \{1,\dots , n\}$. For example, we prove that $|A/A|\sim \frac{2\text {Li}_2(3/4)}{\pi ^2}n^2$ for almost all subsets $A\subset \{1,\dots ,n\}$. We also prove that the proportion of visible lattice points in the lattice $A_1\times \cdots \times A_d$, where $A_i$ is taken at random in [1, n] with $\mathbb P(m\in A_i)=\alpha _i$ for any $m\in [1,n]$, is asymptotic to a constant $\mu (\alpha _1,\dots ,\alpha _d)$ that involves the polylogarithm of order d.