The discrepancy of the Champernowne constant
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911960541429760 |
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| author | Becher, Verónica Graus, Nicole |
| author_facet | Becher, Verónica Graus, Nicole |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The discrepancy of the Champernowne constant Becher, Verónica Graus, Nicole Number Theory Discrete Mathematics 11K16 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. |
| title | The discrepancy of the Champernowne constant |
| topic | Number Theory Discrete Mathematics 11K16 |
| url | https://arxiv.org/abs/2407.13114 |