On two problems of Hardy and Mahler

Abstract

It is a classical result of Mahler that for any rational number $α$ > 1 which is not an integer and any real 0 < c < 1, the set of positive integers n such that $α$ n < c n is necessarily finite. Here for any real x, x denotes the distance from its nearest integer. The problem of classifying all real algebraic numbers greater than one exhibiting the above phenomenon was suggested by Mahler. This was solved by a beautiful work of Corvaja and Zannier. On the other hand, for non-zero real numbers $λ$ and $α$ with $α$ > 1, Hardy about a century ago asked ''In what circumstances can it be true that $λ$$α$ n $\rightarrow$ 0 as n $\rightarrow$ $\infty$? '' This question is still open in general. In this note, we study its analogue in the context of the problem of Mahler. We first compare and contrast with what is known visa -vis the original question of Hardy. We then suggest a number of questions that arise as natural consequences of our investigation. Of these questions, we answer one and offer some insight into others.