On the powers of 3/2 and other rational numbers

Let p > q > 1 be two coprime integers. In this paper, we prove several results about subsets of the interval [0, 1) which does or does not contain all the fractional parts { ξ ( p / q ) n }, n = 0, 1, 2, …, for certain non‐zero real number ξ . We show, for instance, that there are no real ξ for which the union of two intervals [8/39, 18/39] ∪ [21/39, 31/39] contains the set { ξ (3/2) n }, n ∈ N . The most important aspect of this result is that the total length of both intervals 20/39 is greater than 1/2: the same result as above for [0, 1/2) would imply that there are no Mahler's Z ‐numbers which the best known unsolved problem in this area. On the other hand, it is shown that there are infinitely many ξ for which { ξ (3/2) n } ∈ (5/48, 43/48) for each integer n ≥ 0. We also give simpler proofs of few recent results in this area. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)