Membership in random ratio sets

TL;DR

This paper derives formulas for the probability that a rational number belongs to a random ratio set formed from a randomly chosen subset of integers, extending previous results and exploring related graph connectivity considerations.

Contribution

It provides a generalized formula for membership probabilities in random ratio sets and links these probabilities to graph connectivity analysis.

Findings

Derived a formula for the probability that a rational number is in the ratio set.

Extended previous results by Cilleruelo and Guijarro-Ordóñez.

Connected membership probabilities to graph component analysis.

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-Ord\'o\~nez. Moreover, we make some considerations about formulas for the probability of the event $\bigvee_{i=1}^k\!\big(q_i \in \mathcal{A} /\! \mathcal{A}\big)$, 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 $1 \in \mathcal{E}$.