On a problem of W. J. LeVeque concerning metric diophantine approximation II

Abstract

Abstract. We consider the diophantine approximation problemq 2 where f is a fixed function satisfying suitable assumptions. Suppose that x is randomly chosen in the unit interval. In a series of papers that appeared in earlier issues of this journal, LeVeque raised the question whether or not the central limit theorem holds for the solution set of the above inequality (compare also with some work of Erdős). Here, we are going to extend and solve LeVeque's problem.

Central limit theorems for Diophantine approximants

“…We are not the first to explore Central Limit Theorems for Diophantine approximants. The onedimensional case (m = 1) has been thoroughly investigated by Leveque [12,13], Philipp [14], and Fuchs [6], leading to the following result proved by Fuchs [6]: there exists an explicit σ > 0 such that the counting function Central Limit Theorems in higher dimensions when w 1 = · · · = w m = 1/m have recently been studied Dolgopyat, Fayad and Vinogradov [3]. In this paper, using very different techniques, we establish the following CLT for general exponents w 1 , .…”

Invariance Principles in Metric Diophantine Approximation

“…The central limit theorem contained in Corollary 1 can be viewed as an approximation to the central limit theorem proved by the author in a recent paper [3] for the following sequence of random variables W n (x) := #{ p, q |1 ≤ q ≤ n, q ≡ s(r), p/q is a solution of (3)}. (11) This is getting even more lucid if we point out the following asymptotic expansions for σ and τ 2 .…”

“…Especially notice, that the main term in the asymptotic expansion of σ is, according to [3], the constant belonging to the mean value of (11) and that τ 2 is increasing logarithmically with d which could be seen as explanation for F (n) log F (n) to be the order of magnitude of the variance of (11) (again compare with [3]). …”

“…We shall use the last two lemmas together with ideas of [3] and [8] in order to obtain the following result.…”