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 -