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 $\alpha \in (0, 1)$. Several authors studied combinatorial/number-theoretic objects involving $\mathcal{A}$, including the sum set $\mathcal{A} + \mathcal{A}$, the product set $\mathcal{A}\mathcal{A}$, and the ratio set $\mathcal{A} /\! \mathcal{A}$. Generalizing a previous result of Cilleruelo and Guijarro-Ord\'{o}\~{n}ez, we give a formula for the probability that a rational number $q$ belongs to the ratio set $\mathcal{A} /\! \mathcal{A}$. Moreover, we give some results 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. Finally, we provide some open question for future research.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)