Frequency of a Digit in the Representation of a Number and the Asymptotic Mean Value of the Digits

This paper establishes the conditions for the existence of the asymptotic mean value of ternary digits and identifies an infinite, everywhere dense set of numbers that possess this mean despite lacking a well-defined frequency for their digits.

The Gist

Imagine you have a magical number machine. You feed it a number between 0 and 1, and it spits out an endless string of digits in base 3 (ternary). Instead of just 0s and 1s like a computer, this machine uses 0s, 1s, and 2s.

For example, a number might look like this:
0.120102201...

The paper by Klymchuk, Makarchuk, and Prats'ovytyi asks two big questions about these endless strings:

1. The Frequency Question: If you count how many 0s, 1s, and 2s appear, do they settle into a steady rhythm? (e.g., "Exactly 33% are 0s, 33% are 1s, and 33% are 2s").
2. The Average Question: If you treat the digits as numbers and calculate their average, does that average settle down to a specific value?

Here is the breakdown of their findings using simple analogies.

1. The Two Ways to Measure a Number

Think of the digits in a number as a long line of people waiting in a queue.

• Frequency (The "Demographics" Check): You count the people. Are there equal numbers of people wearing red shirts (0), blue shirts (1), and green shirts (2)? If you look at the first 1,000 people, then the first 1,000,000, does the percentage of red shirts stabilize?

  • The Problem: For some numbers, the crowd is chaotic. One minute you have 90% red shirts, the next minute 10%. The percentage never settles. In math terms, the frequency does not exist.

• Asymptotic Mean (The "Average Weight" Check): Instead of counting colors, you assign a value to each person: Red=0, Blue=1, Green=2. You calculate the average weight of the line.

  • The Surprise: You can have a chaotic crowd where the colors never settle (no frequency), but the average weight is perfectly stable!

2. The Big Discovery: Chaos vs. Order

The authors discovered a fascinating paradox. Usually, we think that if the "colors" (frequencies) are chaotic, the "average" must be chaotic too.

But they proved that is not always true.

They constructed a special type of number where:

• The Frequencies are wild and unpredictable. The ratio of 0s, 1s, and 2s keeps swinging back and forth forever. It never settles.
• The Average is perfectly calm. No matter how far you go in the line, the average value of the digits stays exactly where you want it to be (say, 1.5).

The Analogy:
Imagine a DJ mixing music.

• Frequency is like the ratio of Rock songs to Jazz songs. Sometimes the DJ plays 10 Rock songs in a row, then 10 Jazz songs. The ratio is all over the place.
• The Mean is like the "energy level" of the party. Even if the genres are switching wildly, the DJ can adjust the volume and tempo so that the overall energy of the party stays exactly at a steady 7.5 out of 10.

The paper shows that you can have a party with wild genre-switching (no frequency) but a perfectly steady energy level (a stable mean).

3. The "Normal" Numbers

The paper also talks about "Normal Numbers." These are the "boring" but statistically perfect numbers.

• In a normal number, the digits are perfectly random.
• The frequency of 0s, 1s, and 2s is exactly equal (1/3 each).
• Because the frequencies are stable, the average is also stable.
• Key Finding: If you pick a random number from a hat, it is almost certainly a "Normal Number." The weird numbers where frequencies don't exist are extremely rare (mathematically speaking, they take up "zero space" in the number line), even though they are everywhere in a topological sense.

4. The "Everywhere Dense" Set

The authors found an infinite set of these "weird" numbers (where frequencies don't exist but the average does).

• What does "Everywhere Dense" mean? Imagine the number line is a long road. If you pick any spot on that road, no matter how small the patch of road you look at, you will find one of these "weird" numbers hiding there. They are everywhere, like dust motes in a sunbeam, even though they are technically "rare" in terms of total volume.

Summary of the Paper's Contribution

1. The Link: They showed that if you know the frequency of every digit, you automatically know the average. (If the colors are stable, the average is stable).
2. The Exception: They proved the reverse is not true. You can have a stable average without stable frequencies.
3. The Construction: They built a recipe (an algorithm) to create these specific numbers. You can tell the recipe, "I want a number where the average is 1.2, but the frequencies of 0s, 1s, and 2s should be chaotic," and the recipe will generate it.
4. The Result: There is a massive, infinite, and everywhere-dense collection of numbers that behave chaotically in their composition but perfectly in their average.

In a nutshell: You can have a perfectly balanced scale (the average) even if the individual weights you put on it are jumping around wildly (the frequencies). This paper maps out exactly where and how to find those jumping weights.

Technical Summary

Here is a detailed technical summary of the paper "Frequency of a Digit in the Representation of a Number and the Asymptotic Mean Value of the Digits" by S. O. Klymchuk, O. P. Makarchuk, and M. V. Prats'ovytyi.

1. Problem Statement

The paper investigates the relationship between two fundamental properties of real numbers in their $s$-nary (specifically ternary, $s=3$) representation:

1. Digit Frequency ($\nu_i(x)$): The asymptotic frequency of a specific digit $i$ appearing in the expansion of a number $x$.
2. Asymptotic Mean Value of Digits ($r(x)$): The limit of the arithmetic mean of the first $n$ digits of $x$.

The central problem is to determine the conditions under which these two properties coexist. Specifically, the authors address whether the existence of an asymptotic mean value $r(x)$ implies the existence of individual digit frequencies $\nu_i(x)$, and they seek to construct sets of numbers where the mean exists but the individual frequencies do not.

2. Methodology

The authors employ a combination of analytical number theory, limit analysis, and constructive algorithms:

• Definitions and Relations: They formally define the $s$-nary representation, distinguishing between $s$-nary-rational (two representations) and $s$-nary-irrational (unique representation) numbers. They establish the linear relationship between the mean value and the frequencies:
  $$r(x) = \sum_{i=1}^{s-1} i \cdot \nu_i(x)$$
  (Note: $\nu_0(x)$ is determined by the constraint $\sum \nu_i(x) = 1$).

• System of Equations (for $s=3$): For ternary numbers, the authors analyze the system:

  1. $\nu_0 + \nu_1 + \nu_2 = 1$
  2. $0\cdot\nu_0 + 1\cdot\nu_1 + 2\cdot\nu_2 = r(x)$
     They use this system to prove that for $s=3$, the existence of $r(x)$ and any one frequency implies the existence of the other two. Conversely, if $r(x)$ exists but one frequency does not, the others cannot exist either.

• Constructive Algorithm (Block Method): To construct numbers with a specific mean $r(x) = \theta$ but without existing frequencies, the authors design a number $x^*$ composed of alternating "blocks" of digits 0, 1, and 2.

  • The lengths of these blocks ($a_{k1}, a_{k2}, a_{k3}$) are determined by sequences derived from real parameters $x_1$ and $x_2$.
  • They utilize the integral part function $[y]$ and Lemma 2 (concerning the limit of sums of integral parts) to control the density of digits.
  • By oscillating the parameters $x_1$ and $x_2$ in the construction, they ensure the cumulative frequency of a specific digit (e.g., 0) oscillates and fails to converge, while the weighted sum (the mean) converges to the target $\theta$.

• Topological Analysis: The authors analyze the cardinality and topological properties (density, measure) of the sets of numbers satisfying these conditions.

3. Key Contributions and Results

A. Relationship Between Mean and Frequency

• Lemma 1: Proves that if all digit frequencies exist, the asymptotic mean exists and is a linear combination of those frequencies.
• Theorem 2 (Ternary Case): For $s=3$, the existence of the asymptotic mean $r(x)$ and at least one digit frequency implies the existence of all frequencies. If the mean exists but one frequency does not, then none of the frequencies exist. This establishes a strong dependency for the ternary system that does not necessarily hold for higher $s$ without further constraints.
• Corollary 2: In the ternary system, the set of numbers where frequencies exist is identical to the set where frequencies exist and the mean exists.

B. Existence of "Pathological" Numbers

• Theorem 3: The authors prove the existence of an infinite, everywhere dense, and continual (uncountable) set of numbers $W$ such that:
  • The asymptotic mean value of the digits exists and equals a preassigned $\theta \in (0, 2)$.
  • The frequencies of the individual digits ($\nu_0, \nu_1, \nu_2$) do not exist.
• Construction: They explicitly construct such numbers using a block structure where the proportion of digits oscillates indefinitely, preventing frequency convergence, while the weighted average stabilizes.

C. Properties of the Mean Function $r(x)$

• Theorem 4: The function $r(x)$ (asymptotic mean) has the following properties:
  1. It is defined on an everywhere dense set in $[0, 1]$.
  2. It is undefined on an everywhere dense subset of $[0, 1]$.
  3. Its range is the full interval $[0, s-1]$.
  4. It has exactly one level set of full Lebesgue measure: the set of "normal" numbers where $r(x) = \frac{s-1}{2}$. For all other values of the mean, the set of numbers has Lebesgue measure zero.

4. Significance

This paper makes several significant contributions to the metric theory of numbers and fractal geometry:

1. Clarification of Dependencies: It rigorously clarifies the logical dependency between digit frequencies and the digit mean in the ternary system, showing that for $s=3$, the mean cannot "mask" the non-existence of frequencies unless all frequencies fail to exist simultaneously.
2. Counter-Intuitive Existence: It demonstrates that the asymptotic mean is a more robust property than individual frequencies. One can construct numbers with a perfectly stable average digit value that nonetheless exhibit chaotic, non-converging behavior in the distribution of individual digits.
3. Topological Structure: The finding that the set of numbers with a defined mean but undefined frequencies is everywhere dense and continual is crucial. It implies that such "irregular" numbers are not rare anomalies but are structurally pervasive within the real line, despite having Lebesgue measure zero.
4. Generalization of Besicovitch-Eggleston Sets: The work extends the study of Besicovitch-Eggleston sets (defined by fixed frequencies) to sets defined by fixed means, providing a new perspective on the fractal and topological properties of non-normal numbers.

In summary, the paper establishes that while the asymptotic mean of digits is a well-behaved property for almost all numbers (normal numbers), there exists a rich, dense, and uncountable structure of numbers where the mean is well-defined, yet the underlying digit distribution remains fundamentally unstable.