Membership in random ratio sets

Let A be a random set constructed by picking independently each element of {1, ... , n} with probability alpha is an element of (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 is an element of A}. This generalizes a previous result of Cilleruelo and event Vk Guijarro-Ordonez. Moreover, we make some considerations about formulas for the probability of the (qi is an element of A/A), where q1 , ... , qk are rational numbers, showing that they are related to the i=1 study of the connected components of certain graphs. In particular, we give formulas for the probability that qe is an element of A/A for some e is an element of E, where E is a finite or cofinite set of positive integers with 1 is an element of E. (c) 2022 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.