Abstract
Let f 1 and f 2 be two positive numbers of the field \( K = \mathbb{Q}(\sqrt 5 ) \), and let f n+2 = f n+1 + f n for each n ≥ 1. Let us denote by {x} the fractional part of a real number x. We prove that, for each ξ ∉ K, the inequality {ξf n } > 2/3 holds for infinitely many positive integers n. On the other hand, we prove a result which implies that there is a transcendental number ξ such that {ξf n } < 39/40 for each n ≥ 1. Moreover, it is shown that, for every a > 1, there is an interval of positive numbers that contains uncountably many numbers ξ such that {a n} 6 min 2/(a − 1), (34a 2 − 32a + 7)/(9(2a − 1)2) for each n > 1. Here, the minimum is strictly smaller than 1 for each a > 1. In contrast, by an old result of Weyl, for any a > 1, the sequence {ξa n}, n = 1, 2, ..., is uniformly distributed in [0, 1] (and so everywhere dense in [0, 1]) for almost all real numbers ξ.
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Dubickas, A. An approximation property of lacunary sequences. Isr. J. Math. 170, 95–111 (2009). https://doi.org/10.1007/s11856-009-0021-1
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DOI: https://doi.org/10.1007/s11856-009-0021-1