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arXiv · 2407.13114

The discrepancy of the Champernowne constant

Verónica Becher·Nicole Graus
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Abstract

A number is normal in base $b$ if, in its base $b$ expansion, all blocks of digits of equal length have the same asymptotic frequency. The rate at which a number approaches normality is quantified by the classical notion of discrepancy, which indicates how far the scaling of the number by powers of $b$ is from being equidistributed modulo 1. This rate is known as the discrepancy of a normal number. The Champernowne constant $c_{10} = 0.12345678910111213141516\ldots$ is the most well-known example of a normal number. In 1986, Schiffer provided the discrepancy of numbers in a family that includes the Champernowne constant. His proof relies on exponential sums. Here, we present a discrete and elementary proof specifically for the discrepancy of the Champernowne constant.

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Verónica Becher, Nicole Graus. 2024-07-18. The discrepancy of the Champernowne constant. https://arxiv.org/abs/2407.13114

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