p-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials

Let (xₙ) ₍₀ be a linear recurrence sequence of order k2 satisfying xₙ=a₁x₍-₁+a₂x₍-₂++aₖx₍-₊ for all integers n k, where a₁, , aₖ, x₀, , x₊-₁ Z, with aₖ0. In 2017, Sanna posed an open question to classify primes p for which the quotient set of (xₙ) ₍₀ is dense in Qₚ. In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root, where is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in Qₚ, then the quotient set of (xₙ) ₍ ₀ is dense in Qₚ. In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over Q.