-ADIC QUOTIENT SETS: LINEAR RECURRENCE SEQUENCES

Abstract

AbstractLet $(x_n)_{n\geq 0}$ be a linear recurrence of order $k\geq 2$ satisfying $x_n=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$ for all integers $n\geq k$ , where $a_1,\ldots ,a_k,x_0,\ldots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . Sanna [‘The quotient set of k-generalised Fibonacci numbers is dense in $\mathbb {Q}_p$ ’, Bull. Aust. Math. Soc.96(1) (2017), 24–29] posed the question of classifying primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . We find a sufficient condition for denseness of the quotient set of the kth-order linear recurrence $(x_n)_{n\geq 0}$ satisfying $ x_{n}=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$ for all integers $n\geq k$ with initial values $x_0=\cdots =x_{k-2}=0,x_{k-1}=1$ , where $a_1,\ldots ,a_k\in \mathbb {Z}$ and $a_k=1$ . We show that, given a prime p, there are infinitely many recurrence sequences of order $k\geq 2$ whose quotient sets are not dense in $\mathbb {Q}_p$ . We also study the quotient sets of linear recurrence sequences with coefficients in certain arithmetic and geometric progressions.