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

For every positive integer $n$ and every $\delta \in [0,1]$, let $B(n, \delta)$denotetheprobabilisticmodelinwhicharandomset$A \subseteq {1, \dots, n}$isconstructedbychoosingindependentlyeveryelementof${1, \dots, n}$withprobability$\delta$\mathrm{.}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 $k\ge 2$, where $u_0=0$, $u_1\mathrm{\ne }0$, and $a_1,a_2$ are fixed nonzero integers; and let $\alpha$ and $\beta$, with $\mathrm{\mid }\alpha \mathrm{\mid }\ge \mathrm{\mid }\beta \mathrm{\mid }$, be the two roots of the polynomial $X^2-a_{1}X-a_2$. Also, assume that $\alpha \mathrm{/}\beta$ is not a root of unity. We prove that, as $\delta n / \log n \to +\infty$,forevery$A$in$B(n, \delta)$wehave$$\log \operatorname{lcm} (u_a : a \in A) \sim \frac{\delta\operatorname{Li}_2(1 - \delta)}{1 - \delta} \cdot \frac{3\log!\big|\alpha / !\sqrt{(a_1^2, a_2)}\big|}{\pi^2} \cdot n^2 $$withprobability$1 - o(1)$,where$\operatorname{lcm}$ denotes the lowest common multiple, ${\mathrm{Li}}_2$ 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 $\mathrm{lcm}(A)$ due to Cilleruelo, Ru'{e}, \v{S}arka, and Zumalac'{a}rregui.