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Robert Tichy

Publications and source records attributed to Robert Tichy.

At least 19 recordsLinked to original sources
arXiv2026

Lacunary Series, Nonlinear Functionals and Banach Space Structure

In a previous paper \cite{BT} we studied the asymptotic behavior of $\| \sum_{k=1}^N a_k X_{n_k}\|_p$ for lacunary sequences $(X_{n_k})$ of random variables in $L_p$ and used the result to give a necessary and sufficient condition for the first alternative in the Kadec-Pe{\l}czynski theorem in the case $1\le p<2$. In the present paper we extend this result for nonlinear functionals $f_k (a_1 X_{n_1}, \ldots, a_k X_{n_k})$, establishing a uniform version of the subsequence principle of Aldous \cite{ald}. Moreover, we prove Kadec-Pe{\l}czynski type theorems in Orlicz spaces $L_\psi$.

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arXiv2025

A Marcinkiewicz-Zygmund inequality and the Kadec Pe{\l}czyn\'ski theorem in Orlicz spaces

In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pe{\l}czy\'nski-type result -- originally established by the first and third authors for $L^p$ spaces with $1 \le p < 2$ -- to a broader class of Orlicz spaces defined via Young functions $\psi$ satisfying $x \le \psi(x) \le x^2$.

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arXiv2024

Lacunary sequences in analysis, probability and number theory

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudorandomness, Diophantine equations, and the subsequence principle. In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin's resonance theorem for convergence systems. More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.

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arXiv2023

Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

We study the class of univariate polynomials $β_k(X)$, introduced by Carlitz, with coefficients in the algebraic function field $\mathbb F_q(t)$ over the finite field $\mathbb F_q$ with $q$ elements. It is implicit in the work of Carlitz that these polynomials form a $\mathbb F_q[t]$-module basis of the ring $\text{Int}(\mathbb F_q[t]) = \{f \in \mathbb F_q(t)[X] \mid f(\mathbb F_q[t]) \subseteq \mathbb F_q[t]\}$ of integer-valued polynomials on the polynomial ring $\mathbb F_q[t]$. This stands in close analogy to the famous fact that a $\mathbb Z$-module basis of the ring $\text{Int}(\mathbb Z)$ is given by the binomial polynomials $\binom{X}{k}$. We prove, for $k = q^s$, where $s$ is a non-negative integer, that $β_k$ is irreducible in $\text{Int}(\mathbb F_q[t])$ and that it is even absolutely irreducible, that is, all of its powers $β_k^m$ with $m>0$ factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that $β_k$ is not even irreducible if $k$ is not a power of $q$.

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arXiv2022

On a variant of Pillai's problem with transcendental numbers

In this paper, we study the asymptotic behaviour of the number of solutions $(m, n)\in \mathbb{N}^2$ to the inequality $ | α^n - β^m | \leq x $ when $x$ tends to infinity. Here $α, β$ are given multiplicatively independent complex numbers with $|α| > 1$ and $|β|>1$.

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arXiv2021

On the Shorey-Tijdeman Diophantine equation involving terms of Lucas sequences

Let $r\ge 1$ be an integer and ${\bf U}:=\{U_n\}_{n\ge 0}$ be the Lucas sequence given by $U_0=0,~U_1=1$, and $U_{n+2}=rU_{n+1}+U_n$ for $n\ge 0$. In this paper, we explain how to find all the solutions of the Diophantine equation, $AU_{n}+BU_{m}=CU_{n_1}+DU_{m_1}$, in integers $r\ge 1$, $0\le m<n,~0\le m_1<n_1$, $AU_n\ne CU_{n_1}$, where $A,B,C,D$ are given integers with $A\ne 0,~B\ne 0$, $m,n,m_1,n_1$ are nonnegative integer unknowns and $r$ is also unknown.

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arXiv2020

Integers representable as differences of linear recurrence sequences

Let $\{U_n\}_{n \geq 0}$ and $\{V_m\}_{m \geq 0}$ be two linear recurrence sequences. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as differences $ U_n - V_m$. In particular, the density of such integers is $0$.

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arXiv2018

Measurable Sequences

The paper deals with real valued sequences and its distribution on real line.

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arXiv2018

Constrained Triangulations, Volumes of Polytopes, and Unit Equations

Given a polytope $\mathcal{P}$ in $\mathbb{R}^d$ and a subset $U$ of its vertices, is there a triangulation of $\mathcal{P}$ using $d$-simplices that all contain $U$? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of $\mathcal{P}$. Our proof relates triangulations of $\mathcal{P}$ to triangulations of its "shadow", a projection to a lower-dimensional space determined by $U$. In particular, we obtain a formula relating the volume of $\mathcal{P}$ with the volume of its shadow. This leads to an exact formula for the volume of a polytope arising in the theory of unit equations.

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arXiv2016

On the regularity of primes in arithmetic progressions

We prove that for a positive integer $k$ the primes in certain kinds of intervals can not distribute too 'uniformly' among the reduced residue classes modulo $k$. Hereby, we prove a generalization of a conjecture of Recaman and establish our results in a much more general situation, in particular for prime ideals in number fields.

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arXiv2015

The Kadec-Peł czynski theorem in $L^p$, $1\le p<2$

By a classical result of Kadec and Pełczynski (1962), every normalized weakly null sequence in $L^p$, $p>2$ contains a subsequence equivalent to the unit vector basis of $\ell^2$ or to the unit vector basis of $\ell^p$. In this paper we investigate the case $1\le p<2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem is that the limit random measure $μ$ of the sequence satisfies $\int_{\mathbb{R}} x^2 dμ(x)\in L^{p/2}$.

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arXiv2014

On the system $f(nx)$ and probabilistic number theory

Let $f: {\mathbb R}\to {\mathbb R}$ be a measurable function satisfying \begin{equation*} f(x+1)=f(x), \qquad \int_0^1 f(x)\, dx=0, \qquad \int_0^1 f^2(x)\, dx<\infty. \end{equation*} The asymptotic properties of series $\sum c_k f(kx)$ have been studied extensively in the literature and turned out to be, in general, quite different from those of the trigonometric system. As the theory shows, the behavior of such series is determined by a combination of analytic, probabilistic and number theoretic effects, resulting in highly interesting phenomena not encountered in classical harmonic analysis. In this paper we survey some recent results in the field and prove asymptotic results for the system $\{f(nx), n\ge 1\}$ in the case when the function $f$ is not square integrable.

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arXiv2014

On the law of the iterated logarithm for permuted lacunary sequences

It is known that for any smooth periodic function $f$ the sequence $(f(2^kx))_{k\ge 1}$ behaves like a sequence of i.i.d.\ random variables, for example, it satisfies the central limit theorem and the law of the iterated logarithm. Recently Fukuyama showed that permuting $(f(2^kx))_{k\ge 1}$ can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on $(n_k)_{k\ge 1}$, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of the iterated logarithm for any permutation of the sequence $(f(n_k x))_{k \geq 1}$. A similar result is proved for the discrepancy of the sequence $(\{n_k x\})_{k \geq 1}$, where $\{ \cdot \}$ denotes fractional part.

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arXiv2014

On permutations of lacunary series

It is a well known fact that for periodic measurable $f$ and rapidly increasing $(n_k)_{k \geq 1}$ the sequence $(f(n_kx))_{k\ge 1}$ behaves like a sequence of independent, identically distributed random variables. For example, if $f$ is a periodic Lipschitz function, then $(f(2^kx))_{k\ge 1}$ satisfies the central limit theorem, the law of the iterated logarithm and several further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence remains i.i.d.\ after any permutation of its terms, it is natural to expect that the asymptotic properties of lacunary series are also permutation-invariant. Recently, however, Fukuyama (2009) showed that a rearrangement of the sequence $(f(2^kx))_{k\ge 1}$ can change substantially its asymptotic behavior, a very surprising result. The purpose of the present paper is to investigate this interesting phenomenon in detail and to give necessary and sufficient criteria for the permutation-invariance of the CLT and LIL for $f(n_kx)$.

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arXiv2014

On permutations of Hardy-Littlewood-Pólya sequences

Let ${\cal H}=(q_1, \ldots q_r)$ be a finite set of coprime integers and let $n_1, n_2, \ldots$ denote the multiplicative semigroup generated by $\cal H$ and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory and they have remarkable probabilistic and ergodic properties. For example, the asymptotic properties of the sequence $\{n_kx\}$ are very similar to those of independent, identically distributed random variables; here $\{\cdot \}$ denotes fractional part. However, the behavior of this sequence depends sensitively on the generating elements of $(n_k)$ and the combination of probabilistic and number-theoretic effects results in a unique, highly interesting asymptotic behavior. In particular, the properties of $\{n_kx\}$ are not permutation invariant, in contrast to i.i.d. behavior. The purpose of this paper is to show that $\{n_kx\}$ satisfies a strong independence property ("interlaced mixing"), enabling one to determine the precise asymptotic behavior of permuted sums $S_N (σ)= \sum_{k=1}^N f(n_{σ(k)} x)$. As we will see, the behavior of $S_N(σ)$ still follows that of sums of independent random variables, but its growth speed (depending on $σ$) is given by the classical Gál function of Diophantine approximation theory. Some examples describing the class of possible growth functions are given.

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arXiv2014

Lacunary sequences and permutations

By a classical principle of analysis, sufficiently thin subsequences of general sequences of functions behave like sequences of independent random variables. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis. In contrast to "true" random processes, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change radically after rearrangement. The purpose of this paper is to survey some recent results of the authors on permuted function series. We will see that rearrangement properties of lacunary trigonometric series $\sum (a_k\cos n_kx+b_k \sin n_kx)$ and their nonharmonic analogues $\sum c_k f(n_kx)$ are intimately connected with the number theoretic properties of $(n_k)_{k \geq 1}$ and we will give a complete characterization of permutational invariance in terms of the Diophantine properties of $(n_k)_{k \geq 1}$. We will also see that in a certain statistical sense, permutational invariance is the "typical" behavior of lacunary sequences.

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arXiv2014

On the asymptotic behavior of weakly lacunary series

Let $f$ be a measurable function satisfying $$f(x+1)=f(x), \qquad \int_0^1 f(x) dx=0, \qquad \textrm{Var} ~f < + \infty,$$ and let $(n_k)_{k\ge 1}$ be a sequence of integers satisfying $n_{k+1}/n_k \ge q >1$ $(k=1, 2, \ldots)$. By the classical theory of lacunary series, under suitable Diophantine conditions on $n_k$, $(f(n_kx))_{k\ge 1}$ satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences $(n_k)_{k\ge 1}$ as well, but as Fukuyama (2009) showed, the behavior of $f(n_kx)$ is generally not permutation-invariant, e.g. a rearrangement of the sequence can ruin the CLT and LIL. In this paper we construct an infinite order Diophantine condition implying the permutation-invariant CLT and LIL without any growth conditions on $(n_k)_{k\ge 1}$ and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if $f(x)=\sin 2πx$ and $(n_k)_{k\ge 1}$ grows almost exponentially. Finally we prove that, in a suitable statistical sense, for almost all sequences $(n_k)_{k\ge 1}$ growing faster than polynomially, $(f(n_kx))_{k\ge 1}$ has permutation-invariant behavior.

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