On normal numbers in fractals

Abstract: Let $K$ be the ternary Cantor set, and let $\mu$ be the Cantor–Lebesgue measure on $K$. It is well known that every point in $K$ is not 3-normal. However, if we take any natural number $p \ge 2$ that is not a power of 3, then $\mu$-almost every point in $K$ is $p$-normal. This classical result is due to Cassels and W. Schmidt.

Another way to obtain normal numbers from K is by rescaling and translating $K$, then examining the transformed set. A recent nice result by Dayan, Ganguly, and Barak Weiss shows that for any irrational number $t$, for $\mu$-almost all $x \in K$, the product $tx$ is 3-normal.

In this talk, we will discuss these results and their generalizations, including replacing $p$ with an arbitrary beta number and considering more general times-3 invariant measures instead of the Cantor–Lebesgue measure.

dynamical systemsnumber theory

Audience: researchers in the topic

( slides )

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One World Numeration seminar

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

For questions or subscribing to the mailing list, contact the organisers at numeration@irif.fr

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