Topological, metric and fractal properties of the set of real numbers with a given asymptotic mean of digits in their $4$-adic representation in the case when the digit frequencies exist

This paper investigates the topological, metric, and fractal properties of the set of real numbers whose 4-adic digits possess existing frequencies and a specific asymptotic mean, providing an algorithm for constructing such points and establishing conditions for their Lebesgue measure and Hausdorff dimension.

The Gist

Imagine you have a magical, infinite tape recorder. Every real number between 0 and 1 (like 0.5, 0.12345..., or $\pi$ minus 3) can be written as a long, never-ending string of digits in a specific language.

In our daily lives, we use Base-10 (digits 0–9). But this paper looks at numbers written in Base-4 (digits 0, 1, 2, and 3). Think of it like a four-color palette instead of a ten-color one.

The authors, Pratsiovytyi and Klymchuk, are asking a very specific question about these infinite strings: What happens if we look at the "average" color of the digits?

The Core Concept: The "Average Color"

Imagine you are painting a wall using only four colors: Red (0), Blue (1), Green (2), and Yellow (3).

• If you paint the wall with a pattern that repeats "Red, Red, Red...", the average color is Red.
• If you paint "Red, Blue, Green, Yellow" over and over, the average is a muddy mix of all four.
• If you paint randomly, the average settles into a specific shade.

The paper studies the set of all numbers where this "average color" settles on a specific value, let's call it $\theta$ (theta).

• If $\theta = 0$, the number is made almost entirely of 0s.
• If $\theta = 3$, the number is made almost entirely of 3s.
• If $\theta = 1.5$, the number has a mix that averages out to 1.5.

The authors call this the Asymptotic Mean of Digits. It's like asking, "If I listen to this number's song forever, what is the average pitch?"

The Three Types of "Musicians"

The paper divides these numbers into three groups based on how predictable their "song" is:

1. The Predictable Musicians ($\Theta_1$): These numbers have a clear, steady rhythm. If you count how many times each digit (0, 1, 2, 3) appears, the percentage stabilizes. For example, maybe 25% are 0s, 25% are 1s, etc. The paper focuses heavily on this group.
2. The Chaotic Musicians ($\Theta_2$): These numbers are weird. Sometimes the rhythm is steady, sometimes it's not. They might have an average, but the individual frequencies of the digits are messy.
3. The Silent Musicians ($\Theta_3$): These numbers are so chaotic that they don't have a stable average at all. The paper mostly ignores them.

The Big Discoveries (The "Magic" of the Sets)

The authors found some fascinating things about the group of "Predictable Musicians" ($\Theta_1$) who have a specific average $\theta$:

1. The "Everywhere Dense" Property

Imagine you have a jar of sand. If you pick any tiny grain of sand, you can find a grain of "Average-1.5 Sand" right next to it.
The paper proves that these special numbers are everywhere dense. This means if you pick any random number on the number line, you can get infinitely close to it by finding a number with your desired average. They are everywhere, like dust in a sunbeam, even if you can't see them easily.

2. The "Invisible Crowd" (Lebesgue Measure)

Here is the twist:

• If the average is 1.5: The set of these numbers is "huge." In fact, if you picked a random number from a hat, you would almost certainly pick one from this group. It takes up 100% of the space.
• If the average is anything else (like 0 or 3): The set is "invisible" in terms of size. It has zero measure.
  • Analogy: Imagine a library with infinite books. If you look for books written in a specific, rare language, you might find a few. But if you look for books written in a language that only uses the letter "A", you will find a few, but they take up zero space compared to the whole library.
  • So, numbers with an average of 0 or 3 exist, and they are everywhere dense, but they are so "thin" that if you threw a dart at the number line, the chance of hitting one is zero.

3. The Fractal Dimension (The "Roughness" of the Set)

This is the most mathematical part. The authors calculated the Fractal Dimension.

• A straight line has dimension 1.
• A flat sheet has dimension 2.
• A fractal (like a coastline) has a dimension like 1.2 or 1.5. It's "rougher" than a line but "thinner" than a sheet.

The paper shows that for most averages ($\theta \neq 1.5$), the set of numbers forms a fractal. It's a complex, jagged shape that fills space in a weird way.

• For the extreme cases (average 0 or 3), the fractal dimension is 0. This means the set is so sparse it's almost like a collection of isolated points, even though they are everywhere.
• For the middle cases, the dimension is higher, meaning the set is "thicker" and more complex.

How They Built These Numbers (The Recipe)

The paper doesn't just say these numbers exist; it gives a recipe to build them.
Imagine you want to build a number with an average of 1.5.

1. You decide on a target ratio (e.g., 25% zeros, 25% ones, 25% twos, 25% threes).
2. You write a block of digits: 100 zeros, 100 ones, 100 twos, 100 threes.
3. Then you write a much larger block with the same ratio.
4. Then an even larger block.
   By making the blocks get bigger and bigger, the "average" of the whole number settles exactly on your target. The paper proves this recipe always works.

Why Does This Matter?

This might sound like abstract math, but it helps us understand the structure of randomness.

• It tells us how "normal" numbers behave. Most numbers are "normal" (they have an average of 1.5 in base 4).
• It helps us understand the "weird" numbers that break the rules.
• It connects to Cryptography and Data Compression. Understanding how digits distribute helps in creating secure codes or efficient ways to store data.

Summary in One Sentence

The paper maps out the "geography" of numbers based on their average digit value, proving that while numbers with extreme averages are everywhere, they are invisible to the naked eye (zero size), while numbers with a "normal" average form a massive, complex fractal landscape that fills the entire number line.

Technical Summary

Here is a detailed technical summary of the paper "Topological, Metric and Fractal Properties of the Set of Real Numbers with a Given Asymptotic Mean of Digits in Their 4–Adic Representation in the Case When the Digit Frequencies Exist" by M. V. Pratsiovytyi and S. O. Klymchuk.

1. Problem Statement

The paper investigates the structural properties of subsets of real numbers within the interval $[0, 1]$ defined by the asymptotic mean of their 4-adic digits.

• Context: For a real number $x \in [0, 1]$, let its 4-adic representation be $x = \sum_{k=1}^{\infty} \alpha_k(x) 4^{-k}$, where $\alpha_k(x) \in \{0, 1, 2, 3\}$.
• Key Definitions:
  • Digit Frequency ($\nu_i(x)$): The limit of the relative frequency of digit $i$ in the first $n$ positions as $n \to \infty$.
  • Asymptotic Mean ($r(x)$): The limit of the average value of the first $n$ digits:
    $$r(x) = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \alpha_k(x)$$
• Objective: The authors focus on the set $S_\theta = \{x : r(x) = \theta\}$ for a constant $\theta \in [0, 3]$. Specifically, they analyze the subset $\Theta_1 \subset S_\theta$ where all digit frequencies exist (i.e., $\nu_i(x)$ exists for $i=0,1,2,3$).
• Goal: To determine the topological (density, continuity), metric (Lebesgue measure), and fractal (Hausdorff-Besicovitch dimension) properties of $\Theta_1$.

2. Methodology

The authors employ a combination of fractal geometry, ergodic theory, and constructive algorithms.

• Decomposition: The set $S_\theta$ is decomposed into three disjoint subsets based on the existence of digit frequencies. The paper focuses exclusively on $\Theta_1$, where frequencies exist.
• Relation to Besicovitch-Eggleston Sets: The authors establish that $\Theta_1$ is the union of Besicovitch-Eggleston sets $E[\tau_0, \tau_1, \tau_2, \tau_3]$, where $\tau_i$ represents the frequency of digit $i$. The constraint is that the weighted sum of frequencies must equal the target mean $\theta$:
  $$\tau_1 + 2\tau_2 + 3\tau_3 = \theta, \quad \sum \tau_i = 1, \quad \tau_i \geq 0$$
• Optimization: To find the fractal dimension, they utilize the function $f(\tau) = \sum \tau_i \ln \tau_i$ (related to entropy) and apply the Weierstrass theorem to find its minimum over the compact set of valid probability vectors satisfying the mean constraint.
• Constructive Algorithm: A specific algorithm is developed to construct numbers in these sets. This involves creating blocks of digits where the count of each digit $i$ in the $k$-th block is determined by $\lfloor \tau_i s_k \rfloor$, where $s_k$ is a sequence of positive integers growing to infinity. This ensures the asymptotic frequencies converge to the desired $\tau_i$.
• Analytical Tools: The proofs rely on the Stolz-Cesàro theorem for limit calculations, properties of normal numbers, and the standard formula for the Hausdorff dimension of sets defined by digit frequencies.

3. Key Contributions and Results

A. Topological Properties

• Density and Continuity: The set $\Theta_1$ is proven to be everywhere dense in $[0, 1]$ and continuous (closed).
• Special Cases ($\theta = 0, 3$):
  • If $\theta = 0$, the only possible frequency vector is $(1, 0, 0, 0)$. The set is $E[1, 0, 0, 0]$, which is dense but has a Hausdorff dimension of 0.
  • If $\theta = 3$, the only possible frequency vector is $(0, 0, 0, 1)$. The set is $E[0, 0, 0, 1]$, also dense with dimension 0.
  • These sets are described as "anomalously fractal" because they are dense yet have zero dimension.

B. Metric Properties (Lebesgue Measure)

• Zero Measure Case: For any $\theta \neq 1.5$ (i.e., $\theta \neq 3/2$), the set $\Theta_1$ contains no normal numbers. Since the set of normal numbers has full Lebesgue measure, $\Theta_1$ has Lebesgue measure zero for $\theta \neq 1.5$.
• Full Measure Case: For $\theta = 1.5$, the set $\Theta_1$ contains the set of numbers with uniform digit frequencies $(1/4, 1/4, 1/4, 1/4)$. Since this uniform set has full Lebesgue measure, $\Theta_1$ has full Lebesgue measure ($\lambda(\Theta_1) = 1$) when $\theta = 1.5$.

C. Fractal Properties (Hausdorff-Besicovitch Dimension)

• Dimension Estimate: The Hausdorff dimension $\alpha_0(\Theta_1)$ is bounded from below by the entropy of the "most probable" frequency distribution that yields the mean $\theta$.
  $$\alpha_0(\Theta_1) \geq \frac{-m(\theta)}{\ln 4}$$
  where $m(\theta)$ is the minimum value of $\sum \tau_i \ln \tau_i$ subject to the constraints.
• Interpretation: The dimension is strictly positive for $\theta \in (0, 3)$, indicating that while the set may have zero Lebesgue measure (for $\theta \neq 1.5$), it is a "large" set in the sense of fractal geometry.

D. Construction and Uniqueness

• Algorithm: The paper provides a rigorous algorithm to construct a specific number $\hat{x}$ with prescribed frequencies $\tau_i$ and mean $\theta$.
• Uniqueness Theorems: The authors prove that different sequences of block sizes ($s_k$) or different frequency matrices ($p_{in}$) generate distinct real numbers, provided the sequences diverge sufficiently or the frequency differences do not vanish. This confirms the richness of the set's structure.

4. Significance

• Extension of Previous Work: This paper extends the authors' previous research on base-3 representations to base-4. It highlights that as the base $s$ increases ($s > 3$), the set of numbers with a given asymptotic mean exhibits richer and more complex properties compared to lower bases.
• Bridging Concepts: It successfully bridges the concept of digit frequency (a probabilistic/ergodic concept) with asymptotic mean (an arithmetic concept), showing how constraints on the mean restrict the space of possible frequency vectors.
• Fractal Analysis: The results provide precise estimates for the fractal dimension of sets defined by arithmetic constraints on digit sums, contributing to the broader field of fractal geometry and the study of singular distributions.
• Counter-intuitive Results: The finding that the set has full Lebesgue measure only at the specific mean $\theta = 1.5$ (the expected value for a uniform distribution) and zero measure elsewhere, despite being everywhere dense, underscores the delicate nature of "normality" in number theory.

In summary, the paper characterizes the set of real numbers with a fixed 4-adic digit mean, proving it is a dense, closed set with zero Lebesgue measure (except at the mean of 1.5) and providing a lower bound for its fractal dimension based on the entropy of the underlying digit frequencies.