Point Lattices and Oscillating Recurrence Sequences

Abstract

We consider the following question: Which real sequences (a(n)) that satisfy a linear recurrence with constant coefficients are positive for sufficiently large n? We show that the answer is negative for both (a(n)) and (-a(n)), if the dominating characteristic roots in the representation of a(n) as a generalized power sum comprise either two pairs of conjugate complex roots or one negative real root and one pair of conjugate complex roots. The proof uses results from Diophantine approximation and the geometry of numbers. Concerning the case of a real positive dominating root we show what the answer to the question of positivity is for almost all values of the other dominating roots, provided that all dominating roots are simple.