Ratio sets of random sets

We study the typical behaviour of the size of the ratio set A / A for a random subset $$A\subset \{1,\dots , n\}$$A⊂{1,⋯,n}. For example, we prove that $$|A/A|\sim \frac{2\text {Li}_2(3/4)}{\pi ^2}n^2 $$|A/A|∼2Li2(3/4)π2n2 for almost all subsets $$A\subset \{1,\dots ,n\}$$A⊂{1,⋯,n}. We also prove that the proportion of visible lattice points in the lattice $$A_1\times \cdots \times A_d$$A1×⋯×Ad, where $$A_i$$Ai is taken at random in [1, n] with $$\mathbb P(m\in A_i)=\alpha _i$$P(m∈Ai)=αi for any $$m\in [1,n]$$m∈[1,n], is asymptotic to a constant $$\mu (\alpha _1,\dots ,\alpha _d)$$μ(α1,⋯,αd) that involves the polylogarithm of order d.