中文摘要：

任意的无理数x,其无理指数δx∶=sup{δ≥0∶｜x-pq-1｜≤q-2δi.o.pq-1}衡量x可以被有理数逼近的程度.经典的Jarník-Besicovitch定理表明,对于任意的δ≥1,集合{x∈R∶δx≥δ}的Hausdorff维数为δ-1.Barral和Seuret[1]考虑该定理的局部化问题,证明对于任意的连续函数f∶R→[1,＋∞）,集合{x∈R∶δx≥f（x）}的Hausdorff维数为（inf{f（x）∶x∈R}）-1.本文从经典的Jarník-Besicovitch定理出发,利用连分数的理论给出局部Jarník-Besicovitch定理一个简短的证明。

英文摘要：

For any irrational number x,the irrational exponentδx∶=sup{δ≥0∶｜xpq-1｜≤q-2δi.o.pq-1}plays an important role in Diophantine approximation.The classic Jarník-Besicovitch theorem shows that for anyδ≥1,the Hausdorff dimension of the set{x∈R∶δx ≥δ}equalsδ-1.The localization of this theorem is considered by Barral and Seuret[1].They proved that for any continuous function f∶R→[1,＋∞）,the Hausdorff dimension of the set{x∈R∶δx ≥f（x）}is（inf{f（x）∶x∈ R}）-1.In this paper,we give a short proof of the localized Jarník-Besicovitch theorem by continued fraction.