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The discrepancy of the Champernowne constant

Verónica Becher, Nicole Graus

TL;DR

This work provides a discrete, elementary proof of the exact discrepancy for the Champernowne constant $c_{10}$ in base $10$. It establishes matching upper and lower bounds of order $1/\log N$ for the discrepancy $D(c_{10},N)$, showing $D(c_{10},N)=O(1/\log N)$ but not $o(1/\log N)$; in particular, a lower bound with $K=1/(10^3\cdot3)$ holds for infinitely many $N$. The approach relies on careful counting of digit-block occurrences, distinguishing non-overlapping and overlapping occurrences, and decomposing the Champernowne sequence into blocks $s_\ell$ and partial segments $v(N)$. This elementary, combinatorial method provides a contrast to Schiffer's exponential-sum technique and clarifies how the concatenation structure governs the discrepancy. The results extend naturally to other bases $c_b$, with base-dependent constants, highlighting how normality in different bases can display markedly different rates of convergence to uniform distribution.

Abstract

A number is normal in base $b$ if, in its base $b$ expansion, all blocks of digits of equal length have the same asymptotic frequency. The rate at which a number approaches normality is quantified by the classical notion of discrepancy, which indicates how far the scaling of the number by powers of $b$ is from being equidistributed modulo 1. This rate is known as the discrepancy of a normal number. The Champernowne constant $c_{10} = 0.12345678910111213141516\ldots$ is the most well-known example of a normal number. In 1986, Schiffer provided the discrepancy of numbers in a family that includes the Champernowne constant. His proof relies on exponential sums. Here, we present a discrete and elementary proof specifically for the discrepancy of the Champernowne constant.

The discrepancy of the Champernowne constant

TL;DR

This work provides a discrete, elementary proof of the exact discrepancy for the Champernowne constant in base . It establishes matching upper and lower bounds of order for the discrepancy , showing but not ; in particular, a lower bound with holds for infinitely many . The approach relies on careful counting of digit-block occurrences, distinguishing non-overlapping and overlapping occurrences, and decomposing the Champernowne sequence into blocks and partial segments . This elementary, combinatorial method provides a contrast to Schiffer's exponential-sum technique and clarifies how the concatenation structure governs the discrepancy. The results extend naturally to other bases , with base-dependent constants, highlighting how normality in different bases can display markedly different rates of convergence to uniform distribution.

Abstract

A number is normal in base if, in its base expansion, all blocks of digits of equal length have the same asymptotic frequency. The rate at which a number approaches normality is quantified by the classical notion of discrepancy, which indicates how far the scaling of the number by powers of is from being equidistributed modulo 1. This rate is known as the discrepancy of a normal number. The Champernowne constant is the most well-known example of a normal number. In 1986, Schiffer provided the discrepancy of numbers in a family that includes the Champernowne constant. His proof relies on exponential sums. Here, we present a discrete and elementary proof specifically for the discrepancy of the Champernowne constant.
Paper Structure (10 sections, 5 theorems, 114 equations, 1 figure)

This paper contains 10 sections, 5 theorems, 114 equations, 1 figure.

Table of Contents

  1. Normal numbers
  2. Discrepancy estimate of normal numbers
  3. Statement of results
  4. Basic tools
  5. Theorem \ref{['thm:1']}: Upper bound
  6. Counting occurrences
  7. Proof of Theorem \ref{['thm:1']}
  8. Theorem \ref{['thm:2']}: Lower Bound
  9. Witnessing Blocks
  10. Proof of the Theorem \ref{['thm:2']}

Key Result

Proposition 1

A real number $\alpha$ is normal in base $10$ if and only if $\lim_{N \to \infty} D(\alpha, N) = 0$.

Figures (1)

  • Figure 1: Plot of the expansion of the first 250000 digits of Champernowne constant in base $10$, in base $2$ and in base $6$, from left to right. In each base each digit is assigned a different color, and the digits in the expansion are displayed in row-major order.

Theorems & Definitions (33)

  • Definition : Ocurrences counter
  • Definition : Normal number in base $10$
  • Definition : Discrepancy of a sequence
  • Definition : Discrepancy of a number for base $10$
  • Proposition
  • Theorem 1
  • Theorem 2
  • Remark
  • Remark
  • Definition
  • ...and 23 more