Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

TL;DR

This paper introduces the concept of transcendentally normal numbers, proves their abundance, provides explicit constructions, and develops algorithms to compute their digits across all bases, extending to LIL-normal numbers.

Contribution

It defines transcendentally normal numbers, proves their typicality, constructs explicit examples, and creates algorithms for digit computation in all bases, also addressing LIL-normal numbers.

Findings

Almost every real number is transcendentally normal.

Explicit construction of a transcendentally normal number using Sierpinski's method.

Algorithms for computing digits of transcendentally normal and LIL-normal numbers.

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.