On the Distribution of Powers of Real Numbers Modulo 1

Given a strictly increasing sequence of positive real numbers tend- ing to infinity (q n)∞ n=1, and an arbitrary sequence of real numbers (r n)∞ n=1 . We study the set of α ∈ (1, ∞) for which limn→∞αqn − r n= 0. In [3] Dubickas showed that whenever limn→∞(q n+1 − q n) = ∞, there always exists a transcendental α for which limn→∞αqn − r n= 0. Adapting the approach of Bugeaud and Moshchevitin [2], we improve upon this result and show that whenever limn→∞(q n+1 − q n) = ∞, then for any interval I ⊂ (1, ∞) the set of α ∈ I satisfying limn→∞αqn − r n= 0 is of Hausdorff dimension 1.