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 q 2 . Now, we prove it for p > q 2 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 … ] .

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