Membership in random ratio sets

Abstract

Let $\mathcal{A}$ be a random set constructed by picking independently each element of $\{1,\dots ,n\}$ with probability $\alpha \in (0,1)$. We give a formula for the probability that a rational number $q$ belong to the random ratio set $\mathcal{A}/\mathcal{A}:=\{a/b:a,b\in \mathcal{A}\}$. This generalizes a previous result of Cilleruelo and Guijarro-Ordóñez. Moreover, we make some considerations about formulas for the probability of the event ${\bigvee }_{i=1}^k(q_i\in \mathcal{A}/\mathcal{A})$, where $q_1,\dots ,q_k$ 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 $q^e\in \mathcal{A}/\mathcal{A}$ for some $e\in \mathcal{E}$, where $\mathcal{E}$ is a finite or cofinite set of positive integers with $1\in \mathcal{E}$.