On the least common multiple of random $q$-integers

Abstract

For every positive integer $n$ and for every $\alpha \in [0, 1]$, let $\mathcal{B}(n, \alpha)$ denote the probabilistic model in which a random set $\mathcal{A} \subseteq \{1, \dots, n\}$ is constructed by picking independently each element of $\{1, \dots, n\}$ with probability $\alpha$. Cilleruelo, Ru\'{e}, \v{S}arka, and Zumalac\'{a}rregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $\mathcal{A}$. Let $q$ be an indeterminate and let $[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in \mathbb{Z}[q]$ be the $q$-analog of the positive integer $k$. We determine the expected value and the variance of $X := \operatorname{deg} \operatorname{lcm}\!\big([\mathcal{A}]_q\big)$, where $[\mathcal{A}]_q := \big\{[k]_q : k \in \mathcal{A}\big\}$. Then we prove an almost sure asymptotic formula for $X$, which is a $q$-analog of the result of Cilleruelo et al.