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It is proved that a real-valued function $f\left(x\right)=exp\left(\pi i{\chi }_I\left(x\right)\right)$, where I is an interval contained in [0,1), is not of the form $f\left(x\right)=\overline{q\left(2x\right)}q\left(x\right)$ with |q(x)|=1 a.e. if I has dyadic endpoints. A relation of this result to the uniform distribution mod 2 is also shown.

The convergence rate of the expectation of the logarithm of the first return time $R_n$, after being properly normalized, is investigated for ergodic Markov chains. I. Kontoyiannis showed that for any β > 0 we have $log\left[R_n\left(x\right)P_n\left(x\right)\right]=o\left(n^{\beta }\right)$ a.s. for aperiodic cases and A. J. Wyner proved that for any ε >0 we have $-(1+\epsilon )logn\le log\left[R_n\left(x\right)P_n\left(x\right)\right]\le loglogn$ eventually, a.s., where $P_n\left(x\right)$ is the probability of the initial n-block in x. In this paper we prove that $E[log{R}_{(L,S)}-(L-1)h]$ converges to a constant depending only on the process where $R_{(L,S)}$ is the modified first return time with...

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $dₙ\left(x\right):={\sum }_{i=1}^n{1}_E\left(2^{i-1}x\right)\left(mod2\right)$, where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if $N^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that $N^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges a.e. and the limit equals 1/3 or 2/3 depending on x.