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Lower bounds for the Hausdorff dimension of expressible sets

This paper establishes positive lower bounds on the Hausdorff dimension of sets of real numbers defined by specific series expansions, a result that implies certain irrational numbers from Erdős's 1976 construction are not Liouville numbers.

Original authors: Maiken Gravgaard, Simon Kristensen, Jaroslav Hančl

Published 2026-05-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Maiken Gravgaard, Simon Kristensen, Jaroslav Hančl

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 building a very strange, very specific number. You do this by adding up an infinite list of tiny fractions.

The formula looks like this:
$\text{Number} = \frac{1}{d_1 a_1} + \frac{1}{d_2 a_2} + \frac{1}{d_3 a_3} + \dots$

In this paper, the authors are playing with two main ingredients:

The "Base" ( $a_n$ ): These are numbers that get huge, very quickly. Think of them as the denominators of your fractions. They grow so fast that the fractions become microscopic almost instantly.
The "Digits" ( $d_n$ ): These are the numbers you choose to multiply the base by. In the simplest version, you can pick any natural number (1, 2, 3...). But in this paper, the authors restrict you. Maybe you can only pick numbers between 1 and 10, or maybe you can only pick even numbers.

The Big Question: What kind of number do you get?

When you add up these infinite fractions, you get a real number. The authors want to know two things about the collection of all possible numbers you can make with these rules:

Is it "Irrational"? (Can it be written as a simple fraction like 1/2?)
Is it a "Liouville Number"? This is a special, weird type of irrational number. Liouville numbers are "too easy" to approximate with simple fractions. They are so close to rational numbers that they are considered "transcendental" (not the root of any polynomial equation).

For a long time, mathematicians knew that if your "Base" numbers ( $a_n$ ) grew fast enough, the resulting number would be irrational. But there was a catch: if they grew too fast, the number would become a Liouville number.

The Authors' Discovery: The "Gap" Strategy

The authors wanted to find a "sweet spot." They wanted to prove that if you restrict your choices of "Digits" ( $d_n$ ) just right, you can create a set of numbers that are:

Irrational (not simple fractions).
NOT Liouville numbers (they aren't "too easy" to approximate).
Transcendental (mathematically complex).

To do this, they used a clever visual trick involving Cantor Sets.

The Analogy: The Squeezing Sponge
Imagine you have a long stick (representing all possible numbers).

Level 0: You have the whole stick.
Level 1: You cut out the middle parts, leaving only a few smaller sticks.
Level 2: You take those smaller sticks and cut out more middle parts, leaving even tinier sticks.
Repeat forever.

The "Expressible Set" is the dust that remains after you do this forever.

The authors' main job was to prove that when they cut out the middle parts (the "gaps"), the remaining sticks are thick enough to have a non-zero "Hausdorff Dimension."

Hausdorff Dimension is a way to measure the "size" or "roughness" of a shape. A line has dimension 1. A point has dimension 0. A fractal (like the dust in our sponge) might have a dimension of 0.5.
If the dimension is greater than 0, it means the set is "large" in a mathematical sense.
If the dimension is 0, the set is "tiny" (like the set of all Liouville numbers, which is so small it has dimension 0).

The Three Main Results

The paper presents three scenarios based on how fast the "Base" numbers ( $a_n$ ) grow:

1. The Geometric Growth (The "Fast but Steady" Case)
If the base numbers grow like powers of a number (e.g., $2^n, 3^n, 4^n$ ), and you restrict your "digits" to a small, fixed set (like only picking 1, 2, or 3), the authors proved that the resulting set of numbers has a positive dimension.

The Catch: The base must grow fast enough compared to how many digits you are allowed to pick. If you pick too many digits, the "gaps" disappear, and the set becomes a solid block (an interval) rather than a fractal dust.

2. The Double Exponential Growth (The "Explosive" Case)
If the base numbers grow incredibly fast (like $2^{2^n}$ ), the authors showed that even with very specific restrictions on the digits, the resulting set of numbers still has a positive dimension.

Why this matters: This proves that you don't need the numbers to grow infinitely fast to get complex numbers. You can have a "moderate" explosive growth and still get a rich set of numbers that are not Liouville numbers.

3. The "Slower" Double Exponential
They also looked at a slightly slower version of the explosive growth and found similar results, proving that the set of "good" numbers is still substantial.

The "So What?" (Without the Jargon)

The most exciting part of the paper is Corollary 5.

For decades, mathematicians knew that if your base numbers ( $a_n$ ) grew fast enough, the resulting numbers were irrational. But they didn't know if those numbers were "too easy" to approximate (Liouville) or "just right."

The authors proved that Erdős's famous 1976 rule is sharp.

If the base numbers grow too fast, you get Liouville numbers (dimension 0).
But if they grow just a little bit slower (specifically, if the growth rate doesn't explode to infinity too quickly), you get a set of numbers that are not Liouville numbers.

The Bottom Line:
The authors built a mathematical "sieve." By carefully choosing how fast the denominators grow and limiting the choices of the numerators, they proved that you can create a "thick" collection of numbers that are irrational and transcendental, but not the weird, easily-approximated Liouville numbers. They showed that the boundary between "normal" irrational numbers and "weird" Liouville numbers is much more precise than previously thought.

They did not apply this to medicine, engineering, or finance. They simply solved a puzzle about the fundamental nature of numbers and how "big" the set of these special numbers is.

Technical Summary: Lower Bounds for the Hausdorff Dimension of Expressible Sets

Problem Statement
The paper investigates the Hausdorff dimension of "expressible sets," defined as sets of real numbers formed by series of the form:
$x = \sum_{n=1}^{\infty} \frac{1}{d_n a_n}$
where $(a_n)_{n=1}^{\infty}$ is a sequence of real numbers (the "base") and $(d_n)_{n=1}^{\infty}$ is a sequence of natural numbers (the "digits") chosen from specific sets $D_n$ .

The central problem addresses the gap between existing results on the irrationality and transcendence of such sums and the metric properties of the sets they generate. Previous work by Erdős (1975) and Hančl established conditions under which these sums are irrational or Liouville numbers. Specifically, if the sequence $a_n$ grows sufficiently fast (e.g., $\limsup a_n^{1/2^n} = \infty$ ), the resulting numbers are irrational; if the growth is even faster, they are Liouville numbers (a set of Hausdorff dimension zero). However, it was unclear what happens when the growth conditions are intermediate or when restrictions are placed on the digits $d_n$ . The authors aim to establish positive lower bounds for the Hausdorff dimension of these sets, thereby proving that under certain growth conditions, the sets contain non-Liouville (and potentially non-transcendental) numbers.

Methodology
The authors employ a geometric construction approach, modeling the expressible sets as generalized Cantor sets. The strategy involves:

Cantor Set Construction: Viewing the series as a numeration system where $a_n$ acts as the base and $d_n$ as the digits. The set $K(A, (D_n))$ is constructed by iteratively splitting intervals.
Gap Analysis: To apply a known lower bound theorem for the Hausdorff dimension of generalized Cantor sets (Theorem 1, based on Falconer), the authors must ensure that "gaps" exist between the sub-intervals at each level $n$ . This requires proving that the difference between adjacent partial sums is strictly larger than the maximum possible tail sum of the series.
Quantitative Estimates: The authors derive explicit lower bounds for the gap sizes ( $\varepsilon_n$ ) and upper bounds for the number of sub-intervals ( $m_n$ ) at each level. They substitute these estimates into the dimension formula:
$\dim_H(C) \ge \limsup_{n \to \infty} \frac{\log(m_1 \cdots m_n)}{-\log(m_{n+1} \varepsilon_{n+1})}$
Growth Classifications: The analysis is divided into three cases based on the growth rate of the sequence $a_n$ $a_{n}$ :
- Geometric growth ( $a_n = b^n$ ).
- Doubly exponential growth ( $a_n \approx 2^{N^n}$ ).
- Slower doubly exponential growth ( $a_n \approx N^{2^n}$ ).

Key Contributions and Results

Theorem 2 (Geometric Growth): For sequences where $a_n = b^n$ with $b > 4$ and digits restricted to a finite set $D_n = \{1, \dots, K\}$ such that $K^2 < b$ , the authors prove:
$\dim_H(K) \ge \frac{\log(K)}{\log(b)}$
This result demonstrates that restricting digits can reduce the dimension from full (interval) to a positive value, provided the base $b$ is large enough relative to the digit set size.
Theorem 3 (Doubly Exponential Growth): For sequences growing as $a_n = 2^{N^n f(n)}$ with specific constraints on the digit sets $D_n$ , the authors establish a lower bound:
$\dim_H(K) \ge \frac{s}{(N-1)(N-s)}$
where $s$ and $N$ are parameters related to the growth rates of the base and the digit sets.
Theorem 4 (Slower Doubly Exponential Growth): For sequences growing as $a_n = N^{2^n f(n)}$ , a lower bound is derived:
$\dim_H(K) \ge \frac{s}{2-s}$
Notably, this bound is independent of the base parameter $N$ , contrasting with Theorem 3 where the bound diminishes as $N$ increases.
Corollary 5 (Sharpness of Erdős' Criterion): The authors apply their results to integer sequences. They show that if $\limsup_{n \to \infty} a_n^{1/t^n} < \infty$ for some $t \in \mathbb{N}$ , the expressible set $K(A)$ contains numbers that are not $U^*$ -numbers (a class containing Liouville numbers). Since the set of $U^*$ -numbers has Hausdorff dimension zero, this implies that the set $K(A)$ contains numbers that are not Liouville numbers. This confirms that Erdős' condition for generating Liouville numbers is sharp; relaxing the growth condition yields sets with positive dimension containing non-Liouville numbers.

Significance and Claims
The paper claims to provide the first positive lower bounds on the Hausdorff dimension of expressible sets with restricted digits. The primary significance lies in demonstrating that the "Liouville" property is not universal for rapidly growing sequences; rather, it depends critically on the specific growth rate and the size of the digit sets.

The authors explicitly state that their results show that Erdős' 1976 construction yields non-Liouville numbers under weaker growth conditions than previously thought. They note that while their bounds are positive, they suspect they are not optimal, as upper bounds for these specific sets have not yet been calculated. The paper concludes by identifying the removal of the restriction $K^2 < b$ in Theorem 2 and the calculation of upper bounds for these sets as open problems. The work does not propose new applications but rather refines the theoretical understanding of the metric properties of numbers defined by specific series expansions.

The Big Question: What kind of number do you get?

The Authors' Discovery: The "Gap" Strategy

The Three Main Results

The "So What?" (Without the Jargon)

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