On the l.c.m. of random terms of binary recurrence sequences

TL;DR

This paper investigates the asymptotic behavior of the logarithm of the least common multiple of random subsets of a binary recurrence sequence, extending previous deterministic results to a probabilistic setting.

Contribution

It provides a probabilistic asymptotic formula for the lcm of random terms in binary recurrence sequences, generalizing prior deterministic findings.

Findings

Asymptotic formula for log lcm with high probability

Extension from deterministic to probabilistic models

Involvement of dilogarithm in the asymptotic expression

Abstract

For every positive integer $n$ and every $\delta \in [0,1]$, let $B(n, \delta)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $\delta$. Moreover, let $(u_k)_{k \geq 0}$ be an integer sequence satisfying $u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$, for every integer $k \geq 2$, where $u_0 = 0$, $u_1 \neq 0$, and $a_1, a_2$ are fixed nonzero integers; and let $\alpha$ and $\beta$, with $|\alpha| \geq |\beta|$, be the two roots of the polynomial $X^2 - a_1 X - a_2$. Also, assume that $\alpha / \beta$ is not a root of unity. We prove that, as $\delta n / \log n \to +\infty$, for every $A$ in $B(n, \delta)$ we have $$\log \operatorname{lcm} (u_a : a \in A) \sim \frac{\delta\operatorname{Li}_2(1 - \delta)}{1 - \delta} \cdot \frac{3\log\!\big|\alpha / \!\sqrt{(a_1^2, a_2)}\big|}{\pi^2} \cdot n^2 $$ with probability $1 - o(1)$, where $\operatorname{lcm}$ denotes the lowest common multiple, $\operatorname{Li}_2$ is the dilogarithm, and the factor involving $\delta$ is meant to be equal to $1$ when $\delta = 1$. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and M\'aty\'as, who studied the deterministic case $\delta = 1$, and is motivated by an asymptotic formula for $\operatorname{lcm}(A)$ due to Cilleruelo, Ru\'{e}, \v{S}arka, and Zumalac\'{a}rregui.