TY - JOUR

T1 - On a problem of W. J. LeVeque concerning metric diophantine approximation II

AU - Fuchs, Michael

PY - 2004/7/1

Y1 - 2004/7/1

N2 - In an earlier paper, we solved LeVeque's problem, to establish a central limit theorem for the number of solutions of the diophantine inequality |x-p/q| ≤ f(log q)/q2 in unknowns p, q with q > 0, where f is a function satisfying special assumptions and x is chosen randomly in the unit interval. Here, we are interested in the almost sure behavior of the solution set. In particular, we obtain a generalized law of the iterated logarithm and we prove a result that gives strong evidence that the law of the iterated logarithm with the standard norming sequence (suggested by the central limit theorem) holds as well. Both results have to be compared with a theorem of W. M. Schmidt; they imply an inverse to Schmidt's theorem and a strong law of large numbers with an error term that is essentially better than the one provided by Schmidt's result.

AB - In an earlier paper, we solved LeVeque's problem, to establish a central limit theorem for the number of solutions of the diophantine inequality |x-p/q| ≤ f(log q)/q2 in unknowns p, q with q > 0, where f is a function satisfying special assumptions and x is chosen randomly in the unit interval. Here, we are interested in the almost sure behavior of the solution set. In particular, we obtain a generalized law of the iterated logarithm and we prove a result that gives strong evidence that the law of the iterated logarithm with the standard norming sequence (suggested by the central limit theorem) holds as well. Both results have to be compared with a theorem of W. M. Schmidt; they imply an inverse to Schmidt's theorem and a strong law of large numbers with an error term that is essentially better than the one provided by Schmidt's result.

UR - http://www.scopus.com/inward/record.url?scp=3142772246&partnerID=8YFLogxK

U2 - 10.1017/S0305004104007650

DO - 10.1017/S0305004104007650

M3 - Article

AN - SCOPUS:3142772246

VL - 137

SP - 17

EP - 41

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 1

ER -