Numbers Whose Powers Are Arbitrarily Close to Integers

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