On the distribution of digits in dyadic expansions (Q1384124)

This paper provides a precise estimate of the Hausdorff dimension of the following set \[ \Biggl\{x\in [0,1]: \lim_{N\to\infty} {1\over N} \sum^N_{n= 1} \prod^r_{k= 0} \varepsilon_{n+k}(x)= q\Biggr\}, \] where the digits \(\varepsilon_n(x)\in \{0,1\}\) are the digits of the dyadic expansion of \(x\), \(q\in [0,1]\) and \(r= 2,3,4\dots\)\ . The proof involves the construction of Markov measures. The paper ends with a multifractal analysis of these measures.