An approximation property of lacunary sequences

Abstract

Let f 1 and f 2 be two positive numbers of the field K = Q( √ 5), and let f n+2 = f n+1 + fn 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 {ξfn} > 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 {ξfn} < 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 {ξa n } 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.

An Approximation by Lacunary Sequence of Vectors

“…See [5] for a result concerning τ(M) in the range M < 1. (See also [2] and [6] for some related work. )…”

The lonely runner problem for many runners

“…We remark that, by Lemma 6 in [10], for any positive numbers v 1 < · · · < v n and any ε > 0 and T > 0 there is an interval I = [u 0 , u 0 + ε/2v n ], where u 0 > T, such that ||v i t|| < ε for each t ∈ I and each i = 1, . .…”