Fractional parts of powers of large rational numbers

Abstract

It is known that for any real number ξ≠0 and any coprime integers p>q>1 the fractional parts {ξ(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<q2. Now, we prove it for p>q2 as well, but under an additional assumption that ξ≠0 is algebraic. The famous motivating problem in this area is an unsolved Mahler’s conjecture of 1968, which asserts that for ξ≠0 the fractional parts {ξ(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…].