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Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

Chokri Manai

TL;DR

The paper introduces transcendentally normal numbers, defined by the property that $p(x)$ is normal for every non-constant integer-valued polynomial $p$. It develops a unified Sierpiński-cover framework, augmented by stretch functions, to prove full-measure results and to construct explicit t-normal numbers, with an extension to transcendentally LIL-normal numbers and non-asymptotic discrepancy bounds. A key contribution is a computable, digit-by-digit algorithm that produces a transcendentally normal number in all bases simultaneously, and a parallel construction for transcendentally LIL-normal numbers, together with precise non-asymptotic LIL bounds and a discretization argument in the style of Fukuyama–Philipp. The work also establishes structural properties of generalized Sierpiński numbers and discusses open problems in the interaction of normality with algebraic operations. Overall, it provides both existential and constructive results linking normality, polynomial images, and computability, with broad implications for understanding normality under algebraic transformations and for producing explicit computable normal constants.

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.

Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

TL;DR

The paper introduces transcendentally normal numbers, defined by the property that is normal for every non-constant integer-valued polynomial . It develops a unified Sierpiński-cover framework, augmented by stretch functions, to prove full-measure results and to construct explicit t-normal numbers, with an extension to transcendentally LIL-normal numbers and non-asymptotic discrepancy bounds. A key contribution is a computable, digit-by-digit algorithm that produces a transcendentally normal number in all bases simultaneously, and a parallel construction for transcendentally LIL-normal numbers, together with precise non-asymptotic LIL bounds and a discretization argument in the style of Fukuyama–Philipp. The work also establishes structural properties of generalized Sierpiński numbers and discusses open problems in the interaction of normality with algebraic operations. Overall, it provides both existential and constructive results linking normality, polynomial images, and computability, with broad implications for understanding normality under algebraic transformations and for producing explicit computable normal constants.

Abstract

In this work, we study real numbers for which is (absolutely) normal for every non-constant integer-valued polynomial . 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.

Paper Structure

This paper contains 20 sections, 31 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminary Considerations
  3. Nonsingular functions
  4. Stretch functions: general properties
  5. Stretch function for polynomials
  6. Construction of $\mathcal{F}$-normal Numbers
  7. Sierpiński's cover method for normal numbers
  8. Extending Sierpiński's approach to $\mathcal{F}$-normal numbers
  9. Properties of Sierpiński numbers
  10. Construction of $\mathcal{F}^\mathrm{LIL}$-normal numbers
  11. Outline for the construction of $U_b^\mathrm{LIL}(r)$
  12. Non-asymptotic bounds for the standard LIL
  13. A cover for $b$-adic intervals
  14. The Fukuyama-Philipp discretization argument
  15. Computing Transcendentally LIL-Normal Numbers
  16. ...and 5 more sections

Key Result

Theorem 1.1

The set $\Omega_{\infty}$ of transcendentally normal numbers is of full measure. Moreover, the explicit "Sierpiński numbers" $(\xi_{\mathcal{P}}(r))_{r \in (0,1]}$(see eq:sirnumF) are transcendentally normal. The algorithm proposed in Section sec:alg computes a transcendentally normal number $\nu \i

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.4
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 48 more