Fractional parts of powers of large rational numbers

It is known that for any real number xi not equal 0 and any coprime integers p > q > 1 the fractional parts {xi(p/q)(n)), n = 0, 1, 2, ..., cannot all lie in an interval of length strictly smaller than 1/p. It is very likely that they never all belong to an interval of length exactly 1/p. This stronger statement (conjectured by Flatto, Lagarias and Pollington in 1995 and later investigated by Bugeaud) was established by the author in 2009 for q < p < q(2). Now, we prove it for p > q(2) as well, but under an additional assumption that xi not equal 0 is algebraic. The famous motivating problem in this area is an unsolved Mahler's conjecture of 1968, which asserts that for xi not equal 0 the fractional parts (xi(3/2)(n)}, n = 0, 1, 2, ..., cannot all lie in [0, 1/2]. We show that they cannot all lie in [8/57, 805/1539] = [0.14035 ..., 0.52306 ... ]. (C) 2019 Elsevier B.V. All rights reserved.