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Decomposition of real numbers into sums of Lüroth sets

This paper investigates the decomposition of real numbers into sums of Lüroth sets defined by specific digit constraints, establishing results on their congruence modulo 1 and analyzing the Hausdorff dimension of these sets and their sums to extend classical findings from continued fractions.

Original authors: Maiken Gravgaard, Ying Wai Lee

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Maiken Gravgaard, Ying Wai Lee

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 have a giant, infinite library of numbers. In this library, every number is written down not as a standard decimal (like 3.14159...), but as a special code made of a sequence of integers. This is called a Lüroth expansion. Think of it like a unique barcode for every number between 0 and 1.

This paper is a mathematical detective story about what happens when you take two groups of these "coded" numbers and add them together.

Here is the breakdown of the story, using simple analogies:

1. The Two Types of Number Clubs

The authors are interested in two specific "clubs" of numbers based on the rules of their barcodes (the digits in their Lüroth expansion):

The "Small Digit" Club ( $L_{\le k}$ ): These are numbers whose barcodes only use small integers (like 2, 3, 4, up to a limit $k$ ). Imagine a club where you can only wear shoes with sizes 2, 3, or 4.
The "Large Digit" Club ( $L_{\ge k}$ ): These are numbers whose barcodes only use big integers (starting from $k$ and going up forever). Imagine a club where you can only wear shoes size 10 and above.

These clubs are fractals. In math, a fractal is a shape that looks the same no matter how much you zoom in. They are "dusty" sets—they have holes everywhere, and if you tried to measure their length with a ruler, the total length would be zero. They are incredibly sparse.

2. The Big Question: Can We Fill the Gaps?

The central question of the paper is: If we take one number from Club A and add it to a number from Club B, do we get every possible real number?

Think of it like this:

You have a bucket of red sand (Club A) and a bucket of blue sand (Club B). Both buckets are mostly empty space with just a few grains of sand.
If you mix a grain of red sand with a grain of blue sand, do you end up with a solid, continuous wall of purple sand that covers the whole floor? Or do you still have gaps?

In the world of standard decimals (like the famous "Cantor set"), mathematicians already knew that if you add two specific fractal sets together, they can sometimes fill up the entire number line. This paper asks: Does this magic happen with Lüroth expansions too?

3. The Main Discoveries

The authors found that the answer depends entirely on the "rules" of the clubs.

Case A: The "Small Digit" Clubs

They looked at what happens when you add two "Small Digit" clubs together.

The Result: If the limit is low (like only allowing digits 2 and 3), adding two of these sets together does not fill the whole line. There are still gaps. It's like trying to fill a swimming pool with two cups of water; it's not enough.
The Surprise: However, if you allow slightly larger digits (like 2, 3, 4, and 5), or if you add three of the smaller sets together, suddenly the gaps disappear! The sum becomes a solid, continuous interval. It's as if adding just a little bit more variety to the "shoe sizes" allows the sand to pack together perfectly.

Case B: The "Large Digit" Clubs

They looked at clubs where the digits must be huge (e.g., 100, 101, 102...).

The Result: These sets are very sparse. You need to add many of them together to fill the gaps.
The Rule: The authors proved a formula: If you take $k$ $k$ copies of the "Large Digit" club (where the digits start at $k$ $k$ ), and add them all together, you will fill the entire number line.
- Analogy: Imagine you have a very thin, stretched-out rubber band (the set). One band is too thin to cover a gap. But if you stack $k$ of them together, they become thick enough to cover the whole distance.

Case C: Mixing Small and Large

What if you mix a "Small Digit" club with a "Large Digit" club?

The Result: The authors found that if you take a "Small Digit" club (digits up to $k+2$ ) and add it to a "Large Digit" club (digits starting at $k$ ), they perfectly fill the number line. It's a perfect partnership: the small digits fill in the holes left by the large digits, and vice versa.

4. Why Does This Matter?

You might ask, "Who cares about adding weird number codes together?"

The "Fractal" Connection: This connects two different worlds of math: Number Theory (how numbers are built) and Fractal Geometry (shapes that are infinitely complex).
The "Badly Approximable" Mystery: In math, some numbers are "hard to approximate" (they are stubborn and don't look like simple fractions), while others are "easy to approximate." The paper suggests that every real number can be broken down into a sum of a "stubborn" number and an "easy" number. This helps us understand the fundamental structure of the real number line.
The "Magic" of Sums: It shows that even though these sets look like dust (they have zero length), when you combine them, they can become solid blocks. It's a bit like how individual pixels on a screen are just tiny dots, but when you combine millions of them, you get a solid, moving picture.

Summary

This paper is a map showing us exactly how to combine different "types" of numbers to create a solid, unbroken line. It tells us:

Too few rules: You get gaps.
Just the right mix of rules: You get a solid line.
The magic number: There is a specific mathematical threshold where the "dust" suddenly turns into a "solid wall."

The authors didn't just guess; they built a mathematical "construction kit" (using something called thickness and gaps) to prove exactly when and why these number sets merge to fill the universe of real numbers.

1. Problem Statement

The paper investigates the additive structure of Luroth sets, which are subsets of real numbers defined by restrictions on their Luroth expansions. Specifically, the authors study the sumsets of these sets (e.g., $L_A + L_B = \{a+b : a \in L_A, b \in L_B\}$ ) and determine under what conditions these sumsets cover the entire real line modulo 1 (i.e., $L_A + L_B \equiv \mathbb{R} \pmod 1$ ).

The work serves as a direct analogue to classical results in continued fraction theory. In the continued fraction setting, it is known that sets of numbers with bounded partial quotients (e.g., $F_{\le k}$ ) have sumsets that cover $\mathbb{R} \pmod 1$ under specific conditions (Hall's Theorem, Hlavka's extensions). The authors ask: Do similar decomposition theorems hold for Luroth expansions, which are generated by a different dynamical system (the Luroth map) but share fractal properties with continued fractions?

2. Methodology

The authors employ a combination of dynamical systems, fractal geometry, and additive number theory.

General Cantor Set Construction: The authors model Luroth sets ( $L_{\le k}$ , $L_{\ge k}$ , etc.) as General Cantor Sets. They utilize a "Stepwise Complete Construction" (SCC) to define these sets via iterative removal of open intervals. This allows them to apply established theorems regarding the sumsets of Cantor sets.
Thickness and $\gamma$ -values: A central tool is the concept of thickness ( $\tau$ ) and the derived quantity $\gamma = \tau / (1+\tau)$ , introduced by Astels. The authors calculate the exact thickness of Luroth sets by analyzing the ratios of removed intervals to retained intervals in their SCC.
Sumset Theorems: They apply key propositions from the literature (Hall, Hlavka, Astels) which state that if the sum of the $\gamma$ -values of two or more Cantor sets exceeds 1, and certain geometric conditions regarding the sizes of gaps and intervals are met, the sumset contains an interval (or covers $\mathbb{R} \pmod 1$ ).
Hausdorff Dimension Analysis: For dimension results, the authors utilize the theory of Iterated Function Systems (IFS) and the Peres–Shmerkin theorem, which relates the dimension of a sumset to the sum of individual dimensions when the contraction ratios are multiplicatively independent.

3. Key Contributions and Results

A. Congruence Modulo 1 (Additive Decomposition)

The paper establishes several theorems determining when sums of Luroth sets cover $\mathbb{R} \pmod 1$ :

Bounded Above Digits ( $L_{\le k}$ ):
- Theorem 2.1: Proves that $L_{\le 3} + L_{\le 5} \equiv \mathbb{R} \pmod 1$ and $L_{\le 4} + L_{\le 4} \equiv \mathbb{R} \pmod 1$ .
- It also proves $L_{\le 3} + L_{\le 3} + L_{\le 3} \equiv \mathbb{R} \pmod 1$ .
- Conversely, it proves $L_{\le 3} + L_{\le 3} \not\equiv \mathbb{R} \pmod 1$ (the sumset has gaps).
- Conjecture 2.2: The authors conjecture that $L_{\le 3} + L_{\le 4} \equiv \mathbb{R} \pmod 1$ , though this remains unproven.
Bounded Below Digits ( $L_{\ge k}$ ):
- Theorem 2.3: Provides a sufficient condition for the sum of $n$ sets $L_{\ge k_i}$ to cover $\mathbb{R} \pmod 1$ . The condition involves the sum of the diameters of the sets being $\ge 1$ and the sequence of bounds $k_i$ not being too widely spread.
- Corollary 2.4: Proves the analogue of the Cusick-Lee result for continued fractions: $k L_{\ge k} \equiv \mathbb{R} \pmod 1$ for any $k \ge 2$ . This means any real number can be written as the sum of $k$ numbers whose Luroth digits are all $\ge k$ .
Mixed Bounds:
- Theorem 2.5: Proves that $L_{\le k+2} + L_{\ge k} \equiv \mathbb{R} \pmod 1$ .
- Conjecture 2.6: Suggests the stronger statement $L_{\le k} + L_{\ge k} \equiv \mathbb{R} \pmod 1$ .

B. Hausdorff Dimension

The paper analyzes the fractal dimension of these sets and their sums:

Theorem 2.9: For finite sets of digits $A$ and $B$ , the Hausdorff dimension of the sumset is $\dim(L_A + L_B) = \min\{1, \dim L_A + \dim L_B\}$ . This relies on the irrationality of the logarithmic ratios of the contraction factors of the underlying IFS.
Theorem 2.10: Establishes a lower bound for the dimension of $L_{\ge k}$ :
$\dim L_{\ge k} > \frac{1}{2} + \frac{1}{2 \log(\max\{16, k\})}$
Theorem 2.11: Proves that for any $k$ , the sum of two such sets has full dimension: $\dim(L_{\ge k} + L_{\ge k}) = 1$ .
Conjecture 2.12: Proposes that $L_{\ge k} + L_{\ge k}$ actually contains an interval of positive length (not just full dimension).

4. Significance and Impact

Bridging Number Theory and Fractal Geometry: The paper successfully extends classical results from continued fractions (a non-linear dynamical system) to Luroth expansions (a piecewise linear but infinite-segment dynamical system). This demonstrates that the rich additive structure observed in continued fractions is a more general phenomenon applicable to various digit-restriction systems.
New Tools for Fractal Sums: The explicit calculation of thickness ( $\tau$ ) and $\gamma$ -values for Luroth sets provides a concrete framework for analyzing sumsets of general Cantor sets with non-uniform gap structures.
Decomposition of Real Numbers: The results confirm that real numbers can be decomposed into sums of "restricted" Luroth numbers (e.g., numbers with large digits or bounded digits), analogous to decomposing numbers into badly approximable or Liouville numbers in the continued fraction context.
Open Problems: The paper identifies specific gaps in current knowledge (e.g., the $L_{\le 3} + L_{\le 4}$ case) and formulates conjectures that guide future research in the additive combinatorics of fractal sets.

In summary, this work provides a rigorous foundation for understanding the additive properties of Luroth sets, utilizing advanced techniques from fractal geometry to solve problems in Diophantine approximation and number theory.