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arXiv · 2508.09319

Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

Chokri Manai
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Abstract

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.

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Chokri Manai. 2025-08-12. Transcendence Meets Normality: Construction of Transcendentally Normal Numbers. https://arxiv.org/abs/2508.09319

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