实数小数部分的$Q_s$-表示中数字的渐近均值及相关分形几何和分形分析问题

缺失数字分形上的联立与乘法丢番图逼近

Simultaneous and multiplicative Diophantine approximation on missing-digit fractals

本文研究缺失数字分形上的丢番图逼近度量理论，建立了欣钦定理和加拉格尔定理的类比及非齐次推广。

Fractal Subseries of the Harmonic Series

We study the convergence of certain subseries of the harmonic series corresponding to increasing sequences of integers whose digits in a certain base are not uniformly distributed. We also discuss the case of irregular sequences, where the frequency distribution of some of the digits does not exist. Examples are given for irregular sequences where the corresponding harmonic subseries is convergent, or divergent, respectively.

关于分形的度量丢番图逼近

Metric Diophantine approximation on fractals

该研究探讨了缺失数字集中的无理数与多项式分母有理数的逼近问题，并试图证明一个针对大基数缺失数字集的Khintchine型收敛定理。

Singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their s-adic digits

Properties of the set $T_s$ of "particularly non-normal numbers" of the unit interval are studied in details ($T_s$ consists of real numbers $x$, some of whose s-adic digits have the asymptotic frequencies in the nonterminating $s-$ adic expansion of $x$, and some do not). It is proven that the set $T_s$ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ($T_s$ is a superfractal set, i.e., its Hausdorff-Besicovitch dimension is equal to~1). A topological and fractal classification of sets of real numbers via analysi…

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

We consider Farey series of rational numbers in terms of {\it fractal sets} labeled by the Hausdorff dimension with values defined in the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$ and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of {\it dual topological quantum numbers}, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that {\it the universality classes of the quantum Hall transitions are classified in terms of $h$}. The connection between Number Theory and Phy…

贝西科维奇-埃格利斯顿类型的线性分形

Linear fractals of the Besicovitch-Eggleston type

本文研究了具有给定渐近平均数字的区间内数集的拓扑、度量和分形性质，并探讨了其与具有给定数字频率的数之间的联系。