Ratio sets of random sets

TL;DR

This paper investigates the typical size of the ratio set of a random subset of integers and the proportion of visible lattice points in random Cartesian products, revealing asymptotic behaviors and explicit constants.

Contribution

It provides the first asymptotic formulas for the ratio set size of random subsets and the density of visible points in random lattices, involving polylogarithm functions.

Findings

|A/A| asymptotically proportional to n^2 with a constant involving dilogarithm

Proportion of visible lattice points approaches a constant depending on probabilities 

Explicit formulas involving polylogarithms for asymptotic behaviors

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 $d$.