Transcendence Meets Normality: Construction of Transcendentally Normal Numbers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915564016893952 |
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| author | Manai, Chokri |
| author_facet | Manai, Chokri |
| contents | In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transcendence Meets Normality: Construction of Transcendentally Normal Numbers Manai, Chokri Number Theory Probability 11K16, 11Y16, 11J81, 68R15, 60F15 In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems. |
| title | Transcendence Meets Normality: Construction of Transcendentally Normal Numbers |
| topic | Number Theory Probability 11K16, 11Y16, 11J81, 68R15, 60F15 |
| url | https://arxiv.org/abs/2508.09319 |