Ratio sets of random sets

Abstract

We study the typical behavior 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 $d$.