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

Antony, Deepa  ; Barman, Rupam 

Canadian mathematical bulletin, Tome 68 (2025) no. 1, pp. 177-186

Résumé

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$for all integers $n\geq k$, where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$. In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$. In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $, where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$, then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$. In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$.

Détail

DOI : 10.4153/S0008439524000547

Mots-clés : p-Adic number, quotient set, ratio set, linear recurrence sequence