On the discrepancy of Markov-normal sequences (Q679105)

Let the base \(q\) be fixed and \(P=(p_{i,j})_{0\leq i,j\leq q-1}\) be an irreducible Markov transition matrix, \((p_i)_{0\leq i,j\leq q-1}\) the stationary probability vector of \(P\) and \(\mu\) its probability measure. A number \(\alpha\) is said to be Markov normal if in a \(q\)-ary expansion of \(\alpha\) each fixed finite block of digits \(b_0b_1\dots b_k\) appears with an asymptotic frequency of \(p_{b_0}p_{b_0b_1}\dots p_{b_{k-1}b_k}\). The author constructs a Markov normal number \(\alpha\) such that the \(\mu\)-discrepancy of the sequence \((\alpha q^n) \) is \(O(N^{-1/2}\log^2 N)\), where the \(O\)-constant depends only on \(P\) . Until now the best known estimate of the discrepancy was \(O(e^{-c(\log N)^{1/2}})\).