Doubling metric Diophantine approximation in the dynamical system of continued fractions

This paper is concerned with the Diophantine properties of the orbits of real numbers in continued fraction system under the doubling metric. More precisely, let φ be a positive function defined on N. We determine the Lebesgue measure and Hausdorff dimension of the set E(φ)={(x,y)∈[0,1)×[0,1):|Tnx−y|<φ(n)fori.m.n},where T is the Gauss map and “i.m.” stands for “infinitely many”.