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On two functions with complicated local structure
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... In 2013, the investigations of the last section were generalized by the author in several conference abstracts [20,21] and in the paper [22] "One one class of functions with complicated local structure". Consider these results. ...
The present article is devoted to functions from a certain subclass of non-differentiable functions. The arguments and values of considered functions represented by the s-adic representation or the nega-s-adic representation of real numbers. The technique of modeling such functions is the simplest as compared with well-known techniques of modeling non-differentiable functions. In other words, values of these functions are obtained from the s-adic or nega-s-adic representation of the argument by a certain change of digits or combinations of digits.
This paper is devoted to the investigation of the following function $$ f: x=\Delta^{3}_{\alpha_{1}\alpha_{2}...\alpha_{n}...}{\rightarrow} \Delta^{3}_{\varphi(\alpha_{1})\varphi(\alpha_{2})...\varphi(\alpha_{n})...}=f(x)=y, $$ where $\varphi(i)=\frac{-3i^{2}+7i}{2}$, $ i \in N^{0}_{2}=\{0,1,2\}$, and $\Delta^{3}_{\alpha_{1}\alpha_{2}...\alpha_{n}...}$ is the ternary representation of $x \in [0;1]$. That is values of this function are obtained from the ternary representation of the argument by the following change of digits: 0 by 0, 1 by 2, and 2 by 1. This function preserves the ternary digit $0$. Main mapping properties and differential, integral, fractal properties of the function are studied. Equivalent representations by additionally defined auxiliary functions of this function are proved. This paper is the paper translated from Ukrainian (the Ukrainian variant available at https://www.researchgate.net/publication/292970012). In 2012, the Ukrainian variant of this paper was represented by the author in the International Scientific Conference "Asymptotic Methods in the Theory of Differential Equations" dedicated to 80th anniversary of M. I. Shkil (the conference paper available at https://www.researchgate.net/publication/301765319). In 2013, the investigations of the present article were generalized by the author in the paper "One one class of functions with complicated local structure" (https://arxiv.org/pdf/1601.06126.pdf) and in the several conference papers (available at: https://www.researchgate.net/publication/301765326, https://www.researchgate.net/publication/303052308).
We study the Lebesgue structure and fine fractal properties of infinite Bernoulli convolutions, i.e., the distributions of random variables of the form ξ = ∑
k = 1∞
ξ
k
a
k
, where ∑
k = 1∞
a
k
is a convergent positive series and ξ
k
are independent (generally speaking, nonidentically distributed) Bernoulli random variables. Our main aim is to investigate the class of Bernoulli convolutions with essential overlaps generated by a series ∑
k = 1∞
a
k
such that, for any k ∈ ℕ, there exists s
k
∈ ℕ ⋃{0} for which a
k
= a
k+1 =…= \( {a}_{k+{s}_k} \) ≥ \( {r}_{k+{s}_k} \) and, in addition, s
k
> 0 for infinitely many indices k. In this case, almost all (both in a sense of Lebesgue measure and in a sense of fractal dimension) points of the spectrum have continuum many representations of the form ξ = ∑
k = 1∞
ε
k
a
k
with ε
k
∈ {0, 1}. It is shown that the probability measure μ
ξ
has either a pure discrete distribution or a pure singularly continuous distribution. We also establish sufficient conditions for the confidentiality of the family of cylindrical intervals on the spectrum \( {S}_{\mu_{\xi }} \) generated by the distributions of the random variable ξ. In the case of singularity, we also deduce the explicit formula for the Hausdorff dimension of the corresponding probability measure [i.e., the Hausdorff–Besicovitch dimension of the minimal supports of the measure μ
ξ
(in a sense of dimension)].
Decomposing an abelian group into a direct sum of its subsets leads to results that can be applied to a variety of areas, such as number theory, geometry of tilings, coding theory, cryptography, graph theory, and Fourier analysis. Focusing mainly on cyclic groups, Factoring Groups into Subsets explores the factorization theory of abelian groups. The book first shows how to construct new factorizations from old ones. The authors then discuss nonperiodic and periodic factorizations, quasiperiodicity, and the factoring of periodic subsets. They also examine how tiling plays an important role in number theory. The next several chapters cover factorizations of infinite abelian groups; combinatorics, such as Ramsey numbers, Latin squares, and complex Hadamard matrices; and connections with codes, including variable length codes, error correcting codes, and integer codes. The final chapter deals with several classical problems of Fuchs. Encompassing many of the main areas of the factorization theory, this book explores problems in which the underlying factored group is cyclic. .
I present here remarks concerning my work Probabilistic Number Theory, published in 1979/80 as Volumes 239, 240 of the present series. Particular attention is paid to recent developments. References in this supplement that are to those volumes are prefaced by PNT.
