On the uniform distribution modulo 1 of multidimensional LS-sequences
\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}}
scientific article; zbMATH DE number 6363173
scientific article