Numbers Whose Powers Are Arbitrarily Close to Integers - Artūras Dubickas (Axioms, 2025)
In this paper, it is proved that, for any sequence of positive numbers <semantics>ξn</semantics>, <semantics>n=1,2,…</semantics>, which does not converge to zero faster than the exponential function, and any sequence of positive numbers <semantics>δn</semantics>, <semantics>n=1,2,3,…</semantics>, there is an uncountable set of positive numbers S such that, for each <semantics>α>1</semantics> in S, there are infinitely many <semantics>n∈N</semantics> for which the fractional parts <semantics>{ξnαn}</semantics> are smaller than <semantics>δn</semantics>, regardless of how fast the sequence <semantics>δn</semantics> tends to zero. In particular, for any sequence bounded away from zero, namely, <semantics>ξn≥ξ>0</semantics> for <semantics>n≥1</semantics>, it is shown that infinitely many integers n for which the inequality <semantics>{ξnαn}<δn</semantics> is true can be extracted from an arbitrary subsequence <semantics>N</semantics> of positive integers.