On the uniform distribution modulo 1 of multidimensional LS-sequences
type: Item
instance of: scholarly article
title: On the uniform distribution modulo 1 of multidimensional LS-sequences
zbMATH Open document ID: 1323.11049
author: Christoph Aistleitner
author: Markus Hofer
author: Volker Ziegler
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.3470
review text: \textit{S. Kakutani} [Lect. Notes Math. 541, 369--375 (1976; Zbl 0363.60023)] studied the sequence of partitions \(x_{n,1}<x_{n,2}<\dots<x_{n,k(n)}\) of \([0,1]\) obtained in \(n+1\) steps by subdividing every interval \([x_{n,i},x_{n,i+1}]\) of maximal length into two parts, proportional to \(\alpha\) and \(1-\alpha\), respectively. He proved that the sequences of block \(X_n=(x_{n,1},x_{n,2},\dots,x_{n,k(n)})\), \(n=1,2,\dots\), is uniformly distributed for any \(\alpha\in(0,1)\). \textit{I. Carbone} [Ann. Mat. Pura Appl. (4) 191, No. 4, 819--844 (2012; Zbl 1277.11080)] introduced \((L,S)\)-sequences of partitions such that every interval of maximal length in the \(n\)th step is partitioned in the \(n+1\)th step on \(L+S\) intervals such that the first \(L\) intervals have length proportional to \(\beta\) and the remaining \(S\) intervals have length proportional to \(\beta^2\). Thus \(L\beta+S\beta^2=1\) and every interval having either length \(\beta^n\) or \(\beta^{n+1}\). There exists exactly one corresponding number \(\beta\). In the present paper, the authors show that the two-dimensional \((L,S)\)-sequence with one-dimensional components \((L_1,S_1)\) and \((L_2,S_2)\) is not uniformly distributed. More precisely, they prove the following two results: {\parindent=6mm \begin{itemize}\item[(I)] assume that there exist integers \(m\) and \(k\) such that \(\frac{\beta_1^{k+1}}{\beta_2^{m+1}}\in\mathbb Q\) then the two-dimensional \((L,S)\)-sequence is not dense in \([0,1]^2\). \item[(II)] Assume that \(\text{gcd} (L_1,S_1,L_2,S_2)>1\), then the two-dimensional \((L,S)\)-sequence is not dense in \([0,1]^2\). For the proof they use some recurrent relations for the sequences of total number of intervals and the number of long and short intervals in one-dimensional \((L,S)\)-sequence of partitions. \end{itemize}}
reviewed by: Oto Strauch
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11K31
Mathematics Subject Classification ID: 11K38
zbMATH DE Number: 6363173
zbMATH Keywords: uniform distribution
zbMATH Keywords: discrepancy
zbMATH Keywords: LS-sequence
MaRDI profile type: Publication
OpenAlex ID: W1977347398
arXiv ID: 1211.3470
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DOI: 10.1007/S10231-013-0331-0
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description: scientific article; zbMATH DE number 6363173