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

Abstract

For every positive integer n and every δ∈[0,1], let B(n,δ) denote the probabilistic model in which a random set A⊆{1,…,n} is constructed by choosing independently every element of {1,…,n} with probability δ. Moreover, let (uk)k≥0 be an integer sequence satisfying uk=a1uk−1+a2uk−2, for every integer k≥2, where u0=0, u1≠0, and a1,a2 are fixed nonzero integers; and let α and β, with |α|≥|β|, be the two roots of the polynomial X2−a1X−a2. Also, assume that α/β is not a root of unity.We prove that, as δn/log⁡n→+∞, for every A in B(n,δ) we havelog⁡lcm(ua:a∈A)∼δLi2(1−δ)1−δ⋅3log⁡|α/(a12,a2)|π2⋅n2 with probability 1−o(1), where lcm denotes the lowest common multiple, Li2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ=1.This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case δ=1, and is motivated by an asymptotic formula for lcm(A) due to Cilleruelo, Rué, Šarka, and Zumalacárregui.