In , LeVeque proved a central limit theorem for the number of solutions p, q of x-p/q ≤ f(logq)/q2 subject to the conditions 0 < q ≤ n, (p,q) < d, where x ∈ [0,1] and f satisfies certain assumptions. The case d = 1 was considerably improved a few years later by Philipp . We give a common extension of both results by proving almost sure and distribution type invariance principles. Our results entail several corollaries, e.g. a functional central limit theorem and a Strassen's type version of the iterated logarithm law.
- Continued fractions
- Dependent random variables
- Invariance principles
- Metric diophantine approximation