Transcendence meets normality: Construction of transcendentally normal numbers
Chokri Manai
Source abstract
Abstract In this work, we study real numbers for which is (absolutely) normal for every non‐constant polynomial with integer coefficients . 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 Sierpiński's covering method and novel ideas involving stretch functions and Pólya‐type inequalities. 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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