  We consider the problem of deciding $\omega$-regular properties on infinite
traces produced by linear loops. Here we think of a given loop as producing a
single infinite trace that encodes information about the signs of program
variables at each time step. Formally, our main result is a procedure that
inputs a prefix-independent $\omega$-regular property and a sequence of numbers
satisfying a linear recurrence, and determines whether the sign description of
the sequence (obtained by replacing each positive entry with "$+$", each
negative entry with "$-$", and each zero entry with "$0$") satisfies the given
property. Our procedure requires that the recurrence be simple, \ie, that the
update matrix of the underlying loop be diagonalisable. This assumption is
instrumental in proving our key technical lemma: namely that the sign
description of a simple linear recurrence sequence is almost periodic in the
sense of Muchnik, Sem\"enov, and Ushakov. To complement this lemma, we give an
example of a linear recurrence sequence whose sign description fails to be
almost periodic. Generalising from sign descriptions, we also consider the
verification of properties involving semi-algebraic predicates on program
variables.
