Membership in random ratio sets

Abstract

Let A be a random set constructed by picking independently each element of {1,…,n} with probability α∈(0,1). We give a formula for the probability that a rational number q belongs to the random ratio set A/A≔{a/b:a,b∈A}. This generalizes a previous result of Cilleruelo and Guijarro-Ordóñez. Moreover, we make some considerations about formulas for the probability of the event ⋁i=1k(qi∈A/A), where q1,…,qk are rational numbers, showing that they are related to the study of the connected components of certain graphs. In particular, we give formulas for the probability that qe∈A/A for some e∈E, where E is a finite or cofinite set of positive integers with 1∈E.