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Sharp Hausdorff Dimension Bounds for Sets with Bounded and Growing Digits in $N$ -expansions

This paper establishes sharp Hausdorff dimension bounds for sets of irrational numbers in $(0,1)$ with bounded or growing digits in $N$ -expansions, refining classical Jarník-type results and extending Good's theorems to show that the dimension of sets with digits tending to infinity is exactly $1/2$ with explicit dependence on the parameter $N$ .

Original authors: Andreea Catalina Chitu, Gabriela Ileana Sebe, Dan Lascu

Published 2026-03-31

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CC BY 4.0

Original authors: Andreea Catalina Chitu, Gabriela Ileana Sebe, Dan Lascu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe a number, like a secret code, using a special kind of ladder. In math, we usually describe numbers using Continued Fractions. Think of this as a ladder where each rung is a whole number. To get closer to your secret number, you climb up the ladder, rung by rung.

The "rungs" are called digits.

In the standard ladder (Regular Continued Fractions), the rungs can be any number: 1, 2, 3, 100, 1000...
In this paper, the authors are studying a special, modified ladder called an N-expansion. Here, the rules are stricter: every rung must be at least size N. If N is 5, you can't have a rung of size 2; the smallest rung is 5.

The authors are asking a very specific question about these ladders: "How 'thick' or 'complex' are the sets of numbers that follow specific rules about their rungs?"

To measure "thickness" for these weird, jagged sets of numbers, mathematicians use something called Hausdorff Dimension.

A solid line has dimension 1.
A single point has dimension 0.
A "fractal" (like a coastline or a snowflake) has a dimension somewhere in between, like 1.2 or 0.5. It's "thicker" than a point but "thinner" than a full line.

The paper investigates two main scenarios for these N-expansion ladders:

Scenario 1: The "Capped" Ladder (Bounded Digits)

Imagine you are building a ladder, but you are told: "No rung can be bigger than size M."

If M is small, you are forced to use small rungs. The ladder is very restrictive.
If M is huge, you have more freedom.

The Discovery:
The authors figured out exactly how the "thickness" (dimension) of these restricted numbers changes as you change the cap (M) and the minimum rung size (N).

The Analogy: Think of a crowded room. If you tell everyone they must stand within a tiny 1-foot square (very small M), the room feels very empty and structured (low dimension). If you let them stand in a 100-foot square (large M), the room feels much fuller and more chaotic (dimension gets closer to 1).
The Result: They found a precise formula showing that as the cap (M) gets larger, the dimension gets closer to 1, but it does so in a very specific, predictable way that depends on the starting rule (N). They improved upon old formulas to make them more accurate for this specific type of ladder.

Scenario 2: The "Infinite" Ladder (Growing Digits)

Now, imagine a different rule: "The rungs must get bigger and bigger forever."

The first rung might be 10. The next 100. The next 1,000. They keep growing without stopping.
Or, imagine a rule where "Every single rung must be at least size Alpha" (where Alpha is a huge number).

The Discovery:
This is the most surprising part. In the standard ladder (N=1), mathematician I.J. Good proved long ago that if the rungs get infinitely big, the "thickness" of that set of numbers is exactly 0.5. It's like a perfect fractal halfway between a point and a line.

The authors asked: "Does this still hold true for our special N-ladders?"

The Result: Yes! Even though the rules of the ladder changed (the minimum rung size N), the "thickness" of the set where rungs grow forever is still exactly 0.5.
The Nuance: While the final answer (0.5) is the same, the path to get there depends on N. If you set a very high minimum size for the rungs (Alpha), the dimension starts slightly higher than 0.5 and slowly drifts down to 0.5 as the numbers get even bigger. The authors calculated exactly how fast this drift happens.

Why is this important?

Think of the parameter N as the "gravity" of the number system.

In the standard system (N=1), gravity is normal.
In this new system, gravity is stronger (N > 1).

The paper shows that even when you change the "gravity" of the mathematical universe, some fundamental properties of these fractal sets remain surprisingly stable (like the 0.5 dimension for growing digits), while others shift in a very predictable way.

The "How" (The Tools)

To find these answers, the authors used two main tools, which they describe as:

Covering (The Net): Trying to wrap the set of numbers in a net of small intervals to see how much space they take up (giving an upper limit on thickness).
Mass Distribution (The Sand): Pouring "sand" (mathematical weight) onto the set to see how densely it can be packed (giving a lower limit on thickness).

By balancing these two methods, they squeezed the answer until they found the exact "sharp" bounds.

Summary in a Nutshell

The Problem: How complex are numbers that follow strict rules about their "digits" in a special math system?
The "Capped" Rule: If digits are limited to a maximum size, the complexity depends on how big that limit is. The authors gave a better formula for this.
The "Growing" Rule: If digits get infinitely big, the complexity is always exactly 0.5, no matter what the starting rules are.
The Takeaway: Math has hidden symmetries. Even when you change the basic rules of the game (the N-expansion), the deep structure of these number sets remains remarkably consistent.

1. Problem Statement and Context

The paper investigates the metric properties of sets of irrational numbers in the interval $(0, 1)$ defined by restrictions on their digits in $N$ -expansions.

$N$ -expansions: A generalization of regular continued fractions (where $N=1$ ). For an integer $N \ge 1$ , any irrational $x \in (0, 1)$ has a unique expansion:
$x = \cfrac{N}{\varepsilon_1 + \cfrac{N}{\varepsilon_2 + \cfrac{N}{\varepsilon_3 + \dots}}} = [\varepsilon_1, \varepsilon_2, \dots]_N$
where the digits $\varepsilon_n$ belong to the set $\mathbb{N}_N = \{N, N+1, N+2, \dots\}$ .
The Core Question: The authors aim to determine the Hausdorff dimension ( $\dim_H$ $dim_{H}$ ) of two specific families of exceptional sets:
1. Bounded Digits: Sets where digits are restricted to a finite range $[N, M]$ .
2. Growing Digits: Sets where digits tend to infinity or are uniformly large ( $\ge \alpha$ ).

This work extends classical results by Jarník (bounded digits) and Good (growing digits) from the case $N=1$ to general $N$ , providing explicit dependence on the parameter $N$ .

2. Methodology

The authors employ a combination of classical fractal geometry techniques adapted to the specific recurrence relations of $N$ -expansions.

A. Preliminaries and Recurrence Relations

The analysis relies on the fundamental recurrence relations for the convergents $p_n/q_n$ :
$q_n = \varepsilon_n q_{n-1} + N q_{n-2}$
Key estimates are established for the growth of denominators $q_n$ and the lengths of fundamental intervals $I_n(\varepsilon_1, \dots, \varepsilon_n)$ , which are the sets of numbers sharing the first $n$ digits. The length formula is derived as:
$|I_n| = \frac{N^n}{q_n(q_n + q_{n-1})}$
Crucially, the authors establish bounds on the ratio of consecutive interval lengths, which depend on $N$ and the digit values.

B. Upper Bounds (Covering Arguments)

To establish upper bounds for the Hausdorff dimension, the authors use a covering argument:

Construct a cover of the set using fundamental intervals of order $n$ .
Analyze the sum of the $s$ -th powers of the lengths of these intervals ( $\sum |I_n|^s$ ).
Show that if the sum is bounded (or non-increasing) as $n \to \infty$ , then the $s$ -dimensional Hausdorff measure is finite, implying $\dim_H \le s$ .
This involves solving inequalities derived from the telescoping sums of interval lengths.

C. Lower Bounds (Mass Distribution Principle)

To establish lower bounds, the authors use the Mass Distribution Principle:

Construct a probability measure $\mu$ supported on the set of interest.
Define the measure recursively on fundamental intervals such that mass is distributed proportional to $|I_n|^s$ .
Verify the condition $\mu(U) \le c|U|^s$ for small sets $U$ .
This requires proving that the sum of the $s$ -th powers of the lengths of child intervals is greater than or equal to the $s$ -th power of the parent interval length.

3. Key Contributions and Results

Result 1: Generalized Jarník-type Theorem (Bounded Digits)

For the set $E_M = \{x : N \le \varepsilon_n \le M\}$ , the authors provide sharp bounds that refine classical Jarník estimates.

Theorem 1.1: For $M > 2N + 1$ :
$1 - \frac{2(N+1)}{M+1}\frac{1}{\log(N+1)} \le \dim_H(E_M) \le 1 - \frac{N}{(M+1)\log\left(\frac{(M+1)^2}{N}\right)}$
Significance: These bounds explicitly show how the parameter $N$ influences the dimension. As $M \to \infty$ , the dimension approaches 1. The lower bound improves upon classical estimates by incorporating the $N$ -dependence in the logarithmic term.

Result 2: Good-type Theorem (Growing Digits)

For sets where digits grow without bound, the authors extend Good's classical theorem ( $N=1$ ) to general $N$ .

Set $F_N$ : The set of numbers where $\varepsilon_n \to \infty$ .
- Result: $\dim_H(F_N) = 1/2$ .
- Method: Proved using classical covering and mass distribution techniques adapted to $N$ -expansions, offering a self-contained proof distinct from recent Infinite Iterated Function System (IFS) approaches.
Set $F_{\alpha, N}$ : The set of numbers where $\varepsilon_n \ge \alpha$ for all $n$ .
- Theorem 1.2: For sufficiently large $\alpha$ (specifically $\log(\alpha-1) > \exp(1+N)$ ):
  $\frac{1}{2} + \frac{1}{2\log(\alpha+2)} \le \dim_H(F_{\alpha, N}) \le \frac{1}{2} + \frac{\log\log(\alpha-1)}{2\log(\alpha-1)}$
- Significance: The dimension converges to $1/2$ as $\alpha \to \infty$ . The bounds provide precise asymptotic rates of convergence. Remarkably, while the exact solution to the covering equation depends on $N$ , the asymptotic bounds become independent of $N$ for large $\alpha$ , though the validity range of these bounds depends on $N$ .

4. Significance and Implications

Unification of Theory: The paper successfully unifies the metric theory of regular continued fractions ( $N=1$ ) with the broader class of $N$ -expansions, demonstrating how the parameter $N$ acts as a continuous interpolant between different dynamical regimes.
Sharpness of Bounds: The term "sharp" in the title is justified by the fact that the upper and lower bounds converge to the same asymptotic limit ( $1/2$ for growing digits, $1$ for bounded digits) with explicit, optimal error terms involving $N$ .
Methodological Rigor: The authors provide a rigorous, self-contained derivation using elementary covering and mass distribution arguments, avoiding reliance on more abstract IFS frameworks, thereby making the results more accessible to number theorists and fractal geometers.
Future Directions: The paper identifies open problems, including the multifractal analysis of digit growth rates (sets where $\varepsilon_n \sim \psi(n)$ ) and the dimension of exceptional sets in Diophantine approximation for $N$ -expansions, suggesting a rich avenue for future research connecting thermodynamic formalism with dimension theory.

In summary, this work provides a definitive quantitative description of the fractal geometry of $N$ -expansions, refining classical theorems and establishing the precise role of the parameter $N$ in determining the Hausdorff dimension of sets with restricted digits.