Hausdorff Dimensions Of Some Liminf Sets In Diophantine Approximation

Diophantine approximation is an important branch in number theory.In recent years,this branch has made a lot of impressive progress.The content of Diophantine approximation is very rich,and rational approximation to real numbers is the main content of Diophantine approximation.And then there is a close connection between continued fraction expansion and rational approximation to real numbers.The expansion of a continued fraction is always the process of constructing rational approximation solutions,which makes the study on continued fraction to be an important tool in studying rational approximation.This paper is concentrated on studying the Hausdorff dimensions of fraction sets based on the conclusions of F.Adiceam,V.Jarn??k and A.S.Besicovith,which is approximated by rational numbers exactly.And so the conclusion on this is more precise.LetW?= W?(N) = {x ? R : |x- p/q| < q-?, i.o.(p, q) ? N2},W?(Q) = {x ? R : |x- p/q| < q-?i.o.(p, q) ? N × Q}.Let Q be an infinite subset of N.For any ? > 2,this paper consider the Hausdorff dimensions of liminf sets W?\ W?(Q) after F.Adiceam,V.Jarn??k and A.S.Besicovith.The paper shown that if Q is an N \ Q-free set,the Hausdorff dimension of W?\ W?(Q) is the same as that of W?,i.e.2/?.On the other hand,an example is provided indicating that dimHW?\ W?(Q) =1+?(N\Q)?may not hold for general Q when ? ? 2.This paper is divided into four parts.In the first chapter,it introduces the background and the research status of this research and gives current situation and introduces the main conclusions all over the word used in this paper.The second chapter gives some related definitions elementary properties and relevant lemmas,which contain definitions and properties of Hausdorff measure and continued fraction and the mass distribution principle.The last two chapter are the main content of this paper.To show the main conclusions,the paper gives the process of constructing cantor set and lengths of basic cylinders.The conclusion is obtained and the previous conclusions is supplemented.