On the pyatetskii-shapiro normality criterion for continued fractions

Abstract

Analogues of the Pyatetskii-Shapiro normality criterion for continued fractions and for f -expandings with finite initial tiling are established, improving some results by Moshchevitin and Shkredov obtained in the 2002 paper "On Pyatetskii-Shapiro criterion of normality."

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

“…We could use a variant of the "hot spot lemma" (cf. Moshchevitin and Shkredov [18] and Shkredov [30]). However, on the one hand, since their results are for the full shift over finite and infinite alphabets, we need to develop a variant of the "hot spot lemma" for dynamical systems satisfying the specification property.…”

On the joint normality of certain digit expansions

“…In proving Theorems 1.2 and 1.3, we will make use of two techniques that are perhaps not as well known as they should be. The first is the Pyatetskiȋ-Shapiro normality criterion (see [8,13] for some of the original formulations and [1,7,14] for some extensions and improvements).…”