Exact order of discrepancy of normal numbers

Abstract: In the talk we discuss some previous results on the discrepancy of normal numbers and consider the still open question of Korobov: What is the best possible order of discrepancy $D_N$ in $N$, a sequence $(\{b^n\alpha\})_{n\geq 0}$, $b\geq 2,\in\mathbb{N}$, can have for some real number $\alpha$? If $\lim_{N\to\infty} D_N=0$ then $\alpha$ in called normal in base $b$.

So far the best upper bounds for $D_N$ for explicitly known normal numbers in base $2$ are of the form $ND_N\ll\log^2 N$. The first example is due to Levin (1999), which was later generalized by Becher and Carton (2019). In this talk we discuss the recent result in joint work with Gerhard Larcher that guarantees $ND_N\gg \log^2 N$ for Levin's binary normal number. So EITHER $ND_N\ll \log^2N$ is the best possible order for $D_N$ in $N$ of a normal number OR there exist another example of a binary normal number with a better growth of $ND_N$ in $N$. The recent result for Levin's normal number might support the conjecture that $ND_N\ll \log^2N$ is the best order for $D_N$ in $N$ a normal number can obtain.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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