Fractal projections with an application in number theory

Fractal sets of dual topological quantum numbers

The universality classes of the quantum Hall transitions are considered in terms of fractal sets of dual topological quantum numbers filling factors, labelled by a fractal or Hausdorff dimension defined into the interval $1 < h < 2$ and associated with fractal curves. We show that our approach to the fractional quantum Hall effect-FQHE is free of any empirical formula and this characteristic appears as a crucial insight for our understanding of the FQHE. According to our formulation, the FQHE gets a fractal structure from the connection between the filling factors and the Hausdoff dimension of…

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。

Normality preserving operations for Cantor series expansions and associated fractals part II

We investigate how non-zero rational multiplication and rational addition affect normality with respect to $Q$-Cantor series expansions. In particular, we show that there exists a $Q$ such that the set of real numbers which are $Q$-normal but not $Q$-distribution normal, and which still have this property when multiplied and added by rational numbers has full Hausdorff dimension. Moreover, we give such a number that is explicit in the sense that it is computable.

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…

$\widetilde{Q}$-representation of real numbers and fractal probability distributions

A $\widetilde{Q}-$representation of real numbers is introduced as a generalization of the $p-$adic and $Q-$representations. It is shown that the $\widetilde{Q}-$representation may be used as a convenient tool for the construction and study of fractals and sets with complicated local structure. Distributions of random variables $ξ$ with independent $\widetilde{Q}-$symbols are studied in details. Necessary and sufficient conditions for the probability measures $μ_ξ$ associated with $ξ$ to be either absolutely continuous or singular (resp. pure continuous, or pure point) are found in terms of the…

Number theoretic applications of a class of Cantor series fractal functions, II

It is well known that all numbers that are normal of order $k$ in base $b$ are also normal of all orders less than $k$. Another basic fact is that every real number is normal in base $b$ if and only if it is simply normal in base $b^k$ for all $k$. This may be interpreted to mean that a number is normal in base $b$ if and only if all blocks of digits occur with the desired relative frequency along every infinite arithmetic progression. We reinterpret these theorems for the $Q$-Cantor series expansions and show that they are no longer true in a particularly strong way. The main theoretical resu…

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…

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter $\mathbf{r}$ which varies in the $d$-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection …

Fractal in the statistics of Goldbach partition

Some interesting chaos phenomena have been found in the difference of prime numbers. Here we discuss a theme about the sum of two prime numbers, Goldbach conjecture. This conjecture states that any even number could be expressed as the sum of two prime numbers. Goldbach partition r(n) is the number of representations of an even number n as the sum of two primes. This paper analyzes the statistics of series r(n) (n=4,6,8,...). The familiar 3 period oscillations in histogram of difference of consecutive primes appear in r(n).We also find r(n) series could be divided into different levels period …

Jarník集与迪奥菲内分数集合交集的Hausdorff维度

Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals

研究了Jarník集与迪奥菲内分数集合交集的Hausdorff维度，证明其为满维，并指出无理数指数大于2且部分商有界的集合维度为1。

关于整数划分的分形与递推方程

The Fractal and The Recurrence Equations Concerning The Integer Partitions

本文提出利用分形方法解决整数划分计数问题，并借此推导出包括五边形数定理在内的相关递推方程。

Equidistribution from Fractals

We give a fractal-geometric condition for a measure on [0,1] to be supported on points x that are normal in base n, i.e. such that the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes, and it applies also when n is a Pisot number and equidistribution is understood with respect to the beta-map and Parry measure. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent inte…