Metric Properties Of The Set Of Well Approximable Points In Triadic Cantor Set

Diophantine approximation is an important and classic branch in number theory which is devoted to studying approximation of real numbers by rational numbers. In 1842, Dirichlet proved an important theorem about approximation of real numbers by rational numbers for the first time. In 1926, Khintchine's theorem created a method to study the issues related to number theory in the sense of measure, and now we call it as a metric number theory. Jarni first studied a given set of Diophantine approximation points in the sense of Haudorff dimension and Hausdorff measure. And since fractal geometry theory forming, more and more people find the fractal universality. Recent years, diophantine approximation of fractal sets become the integration point of the fractal theory and Diophantine and accept more and more in-depth study.This paper is concentrated on studying metric properties of the set of well approx-imable points in triadic Cantor set, and answeres whether there exist very well approx-imable numbers besides Liouville numbers in triadic Cantor set-an assertion attributed to Mahler.This paper is divided into four parts. We give a brief overview of associated back-ground on diophantine approximation on fractal sets in the preface, and then recall the definition and some properties of Hausdorff measure and dimension. We also cite the mass transference principle a newly developed tools which is much powerful in determin-ing the Hausdorff measure and dimension for limsup sets.The last two sections constitute the main part of this thesis. Given a positive func-tionφ, let A:={3n:n= 0,1,2…}, denote by WA(φ) the set ofφ-well approximation numbers. In the third part, a complete characterization of the 0-∞law in the sense of Hausdorff measure for the set WA(φ) is given. It should be noted is that here we do not requireφis monotonic. Thus, this can be considered as the analogue of Hausdorff measure version of the Duffin-Schaeffer conjecture for WA(φ)∩K.In the last part, we give a simple and direct estimation on Hausdorff dimension of the set of the well approximable points in triadic Cantor set.