Transformations and functions that preserve the asymptotic mean of digits in the ternary representation of a number

This paper investigates transformations on the interval $[0,1)$ and functions that preserve the asymptotic mean of digits in the ternary (or $s$-adic) representation of numbers, establishing the necessary and sufficient conditions for such transformations to belong to this class.

The Gist

Imagine you have a magical number machine. You feed it a number between 0 and 1, and it spits out a long, infinite string of digits in a specific language (like base-3, which uses only 0, 1, and 2).

This paper is about a specific rule for this machine: Can we change the number inside the machine without changing the "average flavor" of the digits it produces?

Here is a breakdown of the paper's ideas using simple analogies.

1. The Setup: The Infinite Bead Necklace

Think of any number between 0 and 1 as an infinite necklace made of beads.

• In our normal world (base-10), the beads are 0 through 9.
• In this paper's world (base-3), the beads are only 0, 1, and 2.

Every number has a unique pattern of these beads. For example, the number 0.1 in base-3 is just a 1 followed by infinite 0s. But most numbers have a chaotic, random-looking mix of 0s, 1s, and 2s.

2. The Two Rules: Frequency vs. Average

The authors are interested in two ways to describe the "flavor" of the necklace:

• Rule A: The Frequency (The Count).
  Imagine you count how many red beads (0s), blue beads (1s), and green beads (2s) are on the necklace. If you look at a very long section, do you see 33% red, 33% blue, and 33% green? This is the digit frequency.
• Rule B: The Asymptotic Mean (The Average Score).
  Imagine you give each bead a score: 0 for a red bead, 1 for a blue bead, and 2 for a green bead. If you add up all the scores and divide by the total number of beads, what is the average score?
  • If you have equal amounts of 0, 1, and 2, the average is 1.
  • If you have mostly 0s, the average is close to 0.
  • If you have mostly 2s, the average is close to 2.

3. The Big Question

The paper asks: Can we rearrange the beads on the necklace so that the Average Score stays exactly the same, but the Frequencies (the specific counts of red, blue, and green) change?

The "Obvious" Answer (The Identity Machine)

If you just leave the necklace alone, the average stays the same, and the frequencies stay the same. This is boring.

The "Tricky" Answer (The Inverter)

Imagine a machine that swaps every 0 with a 2, and every 2 with a 0, but leaves 1s alone.

• Old Average: If you had mostly 0s (low score), you now have mostly 2s (high score). The average changed.
• Conclusion: This machine breaks the rule. It changes the average.

The "Surprising" Discovery (The Main Result)

The authors prove a very strict rule for base-3 numbers:

If you want to keep the Average Score exactly the same, you are forced to keep the Frequencies exactly the same too.

The Analogy:
Imagine you are baking a cake where the "flavor" is determined by the ratio of sugar (0), flour (1), and salt (2).

• The "Average Score" is the total sweetness.
• The "Frequencies" are the exact cups of each ingredient.

The paper says: In this specific kitchen (base-3), if you want the cake to taste exactly the same (same average), you cannot swap the ingredients around. You must use the exact same amount of sugar, flour, and salt.

If you try to change the ratio of ingredients (frequencies) while keeping the total sweetness (average) the same, you will fail. The math proves that for base-3, these two things are locked together. You can't change one without changing the other.

4. The "Almost Everywhere" Loophole

The paper then asks: "What if we don't care about every number, but only the 'normal' ones?"

In math, "normal" numbers are like a perfectly shuffled deck of cards. They have a perfect 1/3 chance of being a 0, 1, or 2.

• The authors found a way to build a machine that works on these "perfectly shuffled" numbers.
• This machine swaps some 1s for 0s and some 1s for 2s in a very specific, rhythmic pattern (like swapping every 7th bead).
• The Result: The machine changes the frequency of the beads (you get slightly more 0s and 2s, and fewer 1s), but because it swaps them in a balanced way, the Average Score stays exactly 1.

The Analogy:
Imagine a crowd of people where everyone is wearing a red, blue, or green hat.

• Strict Rule: If you want the average "color score" to stay the same, you can't change who is wearing what.
• Loophole: If you only look at a specific, huge group of people (the "normal" numbers), you can swap a few hats around in a clever pattern. The crowd looks different (different hat counts), but the average color score remains identical.

5. The "Chaos" Machine

Finally, the authors build a machine that is even weirder.

• It takes a normal number.
• It rearranges the beads in a way that the Average Score is still preserved.
• BUT, it does this by making the frequencies chaotic. The machine ensures that if you count the beads, the percentage of 0s, 1s, and 2s never settles down to a single number. It keeps fluctuating forever.
• Result: The average is stable, but the individual counts are wild and undefined.

Summary

This paper is a detective story about numbers:

1. The Mystery: Can we change the ingredients of a number (frequencies) without changing its average taste?
2. The Verdict: For base-3 numbers, No, unless you are very careful. If you try to change the ingredients, the average taste must change.
3. The Exception: If you only care about "normal" numbers, you can perform a magic trick where you swap ingredients to keep the average taste the same, even though the ingredient counts have changed.
4. The Twist: You can even make the ingredient counts so chaotic they never settle, while the average taste remains perfectly steady.

The authors essentially mapped out the "laws of physics" for how digits behave in these infinite number strings, showing us exactly where the rules are rigid and where there is room for magic.

Technical Summary

Here is a detailed technical summary of the paper "Transformations and Functions That Preserve the Asymptotic Mean of Digits in the Ternary Representation of a Number" by Pratsiovytyi, Klymchuk, and Makarchuk.

1. Problem Statement

The paper investigates the properties of transformations and functions defined on the interval $[0, 1)$ that preserve specific statistical characteristics of the $s$-adic (specifically ternary, $s=3$) representation of real numbers.

The core concepts involved are:

• Digit Frequency ($\nu_i(x)$): The limit of the relative frequency of digit $i$ in the expansion of $x$.
• Asymptotic Mean of Digits ($r(x)$): The limit of the average value of the digits in the expansion, defined as $r(x) = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \alpha_k(x)$.
• The Relationship: For $s=3$, $r(x) = 1\cdot\nu_1(x) + 2\cdot\nu_2(x)$.

The Central Question: The authors seek to determine the necessary and sufficient conditions for a transformation to preserve the asymptotic mean $r(x)$. Specifically, they explore whether preserving the asymptotic mean implies preserving the individual digit frequencies, or if there exist functions that preserve the mean while altering (or destroying) the frequencies.

2. Methodology

The authors employ a combination of measure theory, fractal geometry, and group theory to analyze these transformations.

• Definitions and Group Theory: They define the set of transformations that preserve digit frequencies and prove it forms a non-commutative group under composition.
• Fractal Sets (Besicovitch-Eggleston Sets): They utilize sets $E[\tau_0, \dots, \tau_{s-1}]$ consisting of numbers with specific digit frequencies. They analyze how transformations map these sets and how the Hausdorff-Besicovitch dimension behaves under these mappings.
• Constructive Counterexamples: The authors construct specific, often discontinuous, functions to test the boundaries of their definitions. These constructions involve manipulating the digits of normal numbers (numbers where all digits appear with equal frequency $1/s$) or numbers with specific frequency distributions.
• Asymptotic Analysis: They use limits, Stolz's theorem, and properties of factorials to prove the existence or non-existence of limits for digit frequencies in their constructed examples.

3. Key Contributions and Results

A. Preservation of Digit Frequencies vs. Fractal Dimension

• Result: The authors prove that there exist functions that preserve digit frequencies but do not preserve the Hausdorff-Besicovitch fractal dimension.
• Mechanism: They construct a function $f$ that maps an entire Besicovitch-Eggleston set $E[\tau_0, \tau_1, \tau_2]$ (which has a non-zero fractal dimension) to a single point $x'$. While the digit frequencies of the image point match the original set's parameters, the dimension collapses from a fractal value to 0.
• Significance: This demonstrates that preserving digit frequencies is a weaker condition than preserving fractal dimension.

B. The Equivalence Theorem for Ternary Systems (The Main Theorem)

• Result (Theorem 2): For the ternary system ($s=3$), there exists no function $f: [0, 1) \to [0, 1)$ that preserves the asymptotic mean $r(x)$ everywhere on the set $\Theta$ (where frequencies exist) while failing to preserve the individual digit frequencies.
• Logic:
  • If $r(x) = 0$, the only possible frequency distribution is $\nu_0=1, \nu_1=0, \nu_2=0$.
  • If $r(x) = 2$, the only possible distribution is $\nu_2=1, \nu_0=0, \nu_1=0$.
  • For any $x$ where frequencies exist, the value of $r(x)$ uniquely determines the frequencies in the ternary system. Therefore, if $r(f(x)) = r(x)$, then the frequencies must also be preserved.
• Conclusion: In the ternary system, preserving the asymptotic mean is equivalent to preserving digit frequencies on the set where frequencies exist.

C. Preservation Almost Everywhere (Lebesgue Measure)

• Result (Theorem 3): The authors construct a function $f$ that preserves the asymptotic mean $r(x)$ almost everywhere (on the set of normal numbers $H$) but does not preserve digit frequencies almost everywhere.
• Mechanism: For normal numbers (where $\nu_0=\nu_1=\nu_2=1/3$ and $r(x)=1$), the function $f$ replaces specific occurrences of the digit '1' with '0' and '2' based on a modulo 7 rule.
  • The new frequencies become $\nu_0(f(x)) = 8/21$, $\nu_1(f(x)) = 5/21$, $\nu_2(f(x)) = 8/21$.
  • The new mean is $r(f(x)) = 1(5/21) + 2(8/21) = 21/21 = 1$.
• Significance: This proves that while the equivalence holds "pointwise" on $\Theta$, it breaks down "almost everywhere" because the set of normal numbers allows for a redistribution of frequencies that maintains the weighted sum (the mean) while changing the individual components. The set of such functions has the cardinality of a hypercontinuum.

D. Functions Where Frequencies Do Not Exist

• Result (Section 6): The authors construct a function $f$ such that for any normal number $x$, $r(f(x)) = r(x) = 1$, but the digit frequencies $\nu_i(f(x))$ do not exist for any $i \in \{0, 1, 2\}$.
• Mechanism: The function constructs the image number using blocks of digits with lengths determined by factorials ($n!$) and alternating frequency parameters. By carefully balancing the growth of these blocks, the average converges to 1, but the oscillation of partial sums prevents the individual digit frequencies from converging to a limit.
• Significance: This shows that the asymptotic mean is a more robust invariant than digit frequencies; it can exist even when the frequencies themselves are undefined.

4. Significance and Implications

1. Refinement of Invariant Theory: The paper clarifies the hierarchy of invariants in $s$-adic representations. It establishes that for $s=3$, the asymptotic mean is a "complete" invariant for digit frequencies only when frequencies exist, but becomes a distinct, weaker invariant when considering almost-everywhere properties or non-convergent sequences.
2. Fractal Geometry: The results highlight the disconnect between preserving statistical properties of digits (frequencies/mean) and preserving the geometric complexity (Hausdorff dimension) of the sets they define.
3. Counterintuitive Behavior: The construction of functions that preserve the mean while destroying frequencies (or making them non-existent) challenges the intuition that the mean is simply a linear combination of frequencies. It shows that the limit of the sum can exist even if the limits of the components do not.
4. Cardinality of Function Spaces: The paper demonstrates that the space of functions preserving the asymptotic mean almost everywhere is vast (hypercontinuum cardinality), suggesting a rich structure of such transformations that warrants further study in ergodic theory and dynamical systems.

In summary, the paper provides a rigorous classification of transformations based on their ability to preserve digit statistics, proving a strong equivalence for ternary systems in the strict sense, while revealing significant divergence when considering measure-theoretic "almost everywhere" properties or the existence of limits.