Values of sequences of purely periodic functions

Abstract

Let f: ℝ → ℝ be a nonzero purely periodic function with least period P. For θ (≠ 0) and b both in the interval [0; P), it is shown that when n runs through the nonnegative integers, the nonzero sequence (f(nθ + b)) is purely periodic if θ is a rational multiple of P. While if θ is not a rational multiple of P and f is continuous, the sequence (f(nθ + b)) is dense in the range of f. Moreover, under appropriate conditions, a sequence of the form (∑ d r=1α rf (Pns r/t r + b r)) with rationals s r/t r is shown to be purely periodic with least period being the least common multiple of t 1,...,t d.