The discrepancy of the Champernowne constant
<|reference_start|>The discrepancy of the Champernowne constant: 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.<|reference_end|>
@article{becher2024the,
    title={The discrepancy of the Champernowne constant},
    author={Ver'onica Becher and Nicole Graus},
    journal={arXiv preprint arXiv:2407.13114},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.13114},
    primaryClass={math.NT cs.DM}
}