Doubling metric Diophantine approximation in the dynamical system of continued fractions

摘要

AbstractThis 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 onN. We determine the Lebesgue measure and Hausdorff dimension of the setE(Φ)={(x,y)∈[0,1)×[0,1):|Tnx?y|/mml:mo>Φ(n)fori.m.n},whereTis the Gauss map and “i.m.” stands for “infinitely many”.