On the powers of 3/2 and other rational numbers

Abstract

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.

Powers of Rational Numbers Modulo 1 Lying in Short Intervals

“…Evidently, q(p − 2)/2p(p − q) = 1/p for q = 2. So Theorem 1.3 with q = 2 and t = 0 recovers the above mentioned result (1.1), where p 5, of [6].…”

“…In order to establish the existence of such numbers ξ he used a method of nested intervals. See also Theorem 1.3 of [9] and Theorem 2.2 of [6] for other proofs of the same result. Note that this result is nontrivial only if q − 1 < p − q, i.e., p 2q + 1 (because p = 2q).…”

“…The case of Theorem 1.3 with q odd is already given in Theorem 2.3 (i) of [6]. It can be also proved using the method of nested intervals as in [13] or [4].…”

“…The existence of such ξ was first proved by Akiyama et al [2] for p = 3. Then this was extended to an arbitrary odd integer p 3 by the author [6]. Although the length of this interval I = (−1/p, 1/p) is 2/p, which is greater than 1/(p − 2) for p 5, this result is close to being best possible.…”

“…Another example considered in Theorem 2.3 (iii) of [6] is p/q = 11/4. We proved there that if ( √ 2 + 1)(q − 1) < p < q 2 − q then there is a ξ > 0 such that…”