Equipartition of interval partitions and an application to number theory (Q2750837)

Let \(x\in [0,1)\), \(x=.d_1d_2\cdots\) its decimal expansion and \(x=[0;a_1,a_2,\cdots ]\) its regular continued fraction (RCF) expansion. Let \(y=.d_1d_2\cdots d_n\) and \(z=y+10^{-n}\). Let \(y=[0;b_1, b_2, \ldots , b_l]\) and \(z=[0;c_1, c_2, \ldots , c_k]\) be the RCF expansion of \(y\) and \(z\). Let \(m(n,x)=\max\{i\leq \min(l,k): \text{for all }j\leq i, b_j=c_j\}\). In other words, \(m(n,x)\) is the largest integer such that \(B_n(x) \subset C_{m(n,x)}(x)\), where \(B_n(x)=[y,z]\) is the decimal cylinder of order \(n\) containing \(x\) and \(C_j(x)\) denotes the continued fraction cylinder of order \(j\) containing \(x\). \textit{G. Lochs} [Abh. Math. Semin. Univ. Hamb. 27, 142-144 (1964; Zbl 0124.28003)] has proved the result that for a.e. \(x\in [0,1)\) \(\lim_{n\to \infty}\frac{m(n,x)}n=\frac{6\log 2\log 10}{\pi^2}\). NEWLINENEWLINENEWLINEThe authors give some generalizations of Lochs' result. A finite or countable partition of \([0,1)\) into subintervals is called an interval partition. For an interval partition \(P\) and \(x\in [0,1)\), \(P(x)\) denotes the interval of \(P\) containing \(x\in [0,1)\). Let \(\lambda\) be a Borel probability measure on \([0,1)\), and let \(\mathcal {P}=\{P_n\}_{n=1}^{\infty}\) be a sequence of interval partitions. If \(-\frac{\log\lambda(P_n(x))}n\to c\geq 0\text{ a.e.}(\lambda)\), then we say that \(\mathcal {P}\) has entropy \(c\) a.e. with respect to \(\lambda\). For two sequences of interval partitions \(\mathcal {P}=\{P_n\}_{n=1}^{\infty}\) and \(\mathcal {Q}=\{Q_n\}_{n=1}^{\infty}\), for \(n\in \mathbb{N}\), and for \(x\in [0,1)\), we put \(m_{\mathcal {P},\mathcal {Q}}(n,x)=\sup\{m|P_n(x)\subset Q_m(x)\}\). NEWLINENEWLINENEWLINEOne of the theorems is as follows. Let \(\mathcal {P}=\{P_n\}_{n=1}^{\infty}\) and \(\mathcal {Q}=\{Q_n\}_{n=1}^{\infty}\) be sequences of interval partitions, and let \(\lambda\) be a Borel probability measure on \([0,1)\). Suppose that for some constants \(c>0\) and \(d>0\), \(\mathcal {P}\) has entropy \(c\) a.e. with respect to \(\lambda\) and \(\mathcal {Q}\) has entropy \(d\) a.e. with respect to \(\lambda\). Then \(\frac{m_{\mathcal {P},\mathcal {Q}}(n,x)}{n}\to \frac cd\) a.e.(\(\lambda\)). NEWLINENEWLINENEWLINEAs its application, the authors prove a conjecture in [\textit{W. Bosma, K. Dajani} and \textit{C. Kraaikamp}, Entropy and counting correct digits, University of Nijmegen, Report No.~9925 (1999)]. Furthermore, they also give a convergence in measure result analogous to the previous theorem.