Construction of absolutely normal numbers

Abstract: Let $b\geq2$ be a positive integer. Then every real number $x\in[0,1]$ admits a $b$-adic representation with digits $a_k$. We call the real $x$ simply normal to base $b$ if every digit $d\in\{0,1,\dots,b-1\}$ occurs with the same frequency in the $b$-ary representation. Furthermore we call $x$ normal to base $b$, if it is simply normal with respect to $b$, $b^2$, $b^3$, etc. Finally we call $x$ absolutely normal if it is normal with respect to all bases $b\geq2$.

In the present talk we want to generalize this notion to normality in measure preserving systems like $\beta$-expansions and continued fraction expansions. Then we show constructions of numbers that are (absolutely) normal with respect to several different expansions.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

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