Ratio sets of random sets
Javier Cilleruelo, Jorge Guijarro-Ordonez

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 for a random subset . For example, we prove that for almost all subsets . We also prove that the proportion of visible lattice points in the lattice , where is taken at random in with for any , is asymptotic to a constant that involves the polylogarithm of order .
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