Arithmetic Simulation of Random Processes

On the pyatetskii-shapiro normality criterion for continued fractions

“…Sharp analogues of Theorem 1 were obtained in [8] for some special dynamical systems (including those examined in [11]), for example: for finite Markov chains and for generalized Bernoulli shifts. In the present paper, we obtain a sharp variant of Theorem 1 for dynamical system associated with continued fractions as well as for some other class of dynamical systems with finite initial tiling.…”

“…In [13] Pyatetskii-Shapiro considered the dynamical system generated by the map x → {qx}, q ∈ N, q > 1, on the interval [0, 1) and obtained a criterion for point x 0 to be normal (1). Later some generalizations and improvements of the original criterion from [13] is obtained in several papers by Pyatetskii-Shapiro and Postnikov [11,12,14].…”

Construction of $$\mu $$ μ -normal sequences

“…Normal sequences for Bernoulli shifts and continued fractions were already investigated by Postnikov and Pyateckiȋ [25,26], see also Postnikov [24]. Furthermore normal sequences for Markov shifts and intrinsically ergodic subshifts were constructed by Smorodinsky and Weiss [31].…”

Continuum cardinality of the set of solutions of one class of equations that contain the function of frequency of ternary digits of a number

“…It is easy to give an example of a number for which (i) there exist frequencies of all ternary digits [1], (ii) the frequency of at least one digit does not exist [2,3], (iii) none of the ternary digits has a frequency [4].…”