Let
$(x_n)_{n\geq 0}$
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.