On a problem of Mahler and Szekeres on approximation by roots of integers (Q1001953)

Let \(\alpha\) be a real number greater than \(1\), and define \(\Lambda(\alpha)\) to be the set of limit points of the sequence \(\|\alpha^n\|^{1/n}\), \(n=1,2,3,\dots\) where \(\| x\| \) denotes the distance from \(x\) to the nearest \(z\in\mathbb Z\). In this paper, various results about \(\Lambda(\alpha)\) are proved. When \(\alpha\) is algebraic, \(\Lambda(\alpha)\) is completely characterized (Theorem 1). It consists of at most two points. On the other hand, uncountably many values of \(\alpha\) are found for which \(\Lambda(\alpha)\) is the whole interval \([0,1]\) (Theorem 2). It is also shown that the set of \(\alpha>1\) for which \(\Lambda(\alpha)\) contains \(0\) has Hausdorff dimension \(0\) (Theorem 3). Among other results, it is shown that for any \(\tau>1\) and interval \((a,b)\subset (1,\infty)\) the set \(\{\alpha\in (a,b):\|\alpha^n\|\leq n^{-\tau}\) for infinitely many positive integers \(n\}\) has Lebesgue measure \(0\) and Hausdorff dimension 1 (Theorem 6). Some open questions are discussed at the end of the paper.