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, Rué, Šarka, and Zumalacárregui 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 $X:=\mathrm{deg}\mathrm{lcm}([\mathcal{A}{]}_q)$, where $[\mathcal{A}{]}_q:=\{[k{]}_q:k\in \mathcal{A}\}$. Then we prove an almost sure asymptotic formula for $X$, which is a $q$-analog of the result of Cilleruelo et al.