insight - Scientific Computing - # Hausdorff Dimension of Subrings

Constructing an Uncountable Subring of Real Numbers with Hausdorff Dimension Zero

Q: Can this construction be generalized to create uncountable subrings with other specific Hausdorff dimensions between zero and one?

This is a very interesting question that the paper leaves open for further investigation. While the provided construction specifically yields a subring with Hausdorff dimension zero, generalizing it to achieve other dimensions between zero and one is not immediately obvious. Here's why: Controlling the Growth: The key to achieving Hausdorff dimension zero in the paper is the careful control of the growth rate of the set nS. The bound on gn(z) (the number of elements in nS less than z) ensures that the sets An,t can be covered efficiently by intervals of decreasing size. To achieve a specific Hausdorff dimension d between 0 and 1, one would need to construct sets analogous to nS with a growth rate carefully tuned to d. This would likely involve a deeper understanding of the interplay between additive number theory (how the sums in nS behave) and fractal geometry (how this growth rate translates to Hausdorff dimension). Uncountability and Ring Structure: It's not enough to simply control the Hausdorff dimension. The construction also needs to ensure the resulting set is uncountable and forms a ring. Modifying the construction to target a specific Hausdorff dimension might inadvertently affect these properties. For instance, simply slowing down the growth rate of nS might lead to a countable set, and arbitrarily changing its structure might not result in closure under addition and multiplication. In summary, while generalizing the construction to other Hausdorff dimensions is an intriguing possibility, it would require significant further research. It would demand a delicate balancing act between controlling the set's growth rate to achieve the desired dimension while preserving its uncountability and ring structure.

Q: Could there be an uncountable subring of ℝ with Hausdorff dimension zero that is also a field?

It's highly unlikely. Here's why: Fields and Inverses: The crucial difference between a ring and a field is the existence of multiplicative inverses for all nonzero elements. In a field, if x is a nonzero element, there exists an element x-1 such that x * x-1 = 1. Hausdorff Dimension and Density: Sets with Hausdorff dimension zero are "thin" in the sense that they have Lebesgue measure zero. However, a subfield of ℝ cannot be too thin. If a subfield of ℝ contains an element x arbitrarily close to zero, then it must also contain its inverse x-1, which would be arbitrarily large. This suggests that a subfield cannot be "sparsely" distributed within ℝ like a set of Hausdorff dimension zero. Corollary 1 Implication: Corollary 1 in the paper demonstrates that the constructed ring A only contains dyadic rationals (numbers of the form a/2k). A field would need to contain the multiplicative inverses of all its nonzero elements. However, the multiplicative inverses of most dyadic rationals are not themselves dyadic rationals. For example, the inverse of 1/3 is 3, which is not a dyadic rational. Therefore, while not explicitly proven in the paper, it's highly improbable to have an uncountable subring of ℝ that is both a field and has Hausdorff dimension zero. The requirements of a field seem fundamentally at odds with the "thinness" implied by Hausdorff dimension zero.

Q: How does the existence of such a subring impact our understanding of the "size" and complexity of the real number line?

The existence of such a subring provides a fascinating glimpse into the intricate structure of the real number line, challenging our intuitive notions of "size" and complexity: Beyond Countability: We typically differentiate the "size" of infinite sets using cardinality. The real numbers are uncountable, while the rationals are countable. However, Hausdorff dimension offers a finer measure of size within the realm of uncountable sets. The existence of an uncountable subring with Hausdorff dimension zero reveals that even within the uncountable real numbers, there exist sets of dramatically different "sizes," some far "thinner" than others. Hidden Algebraic Structure: The real numbers are not just a set; they form a field with rich algebraic properties. The constructed subring, while "small" in the sense of Hausdorff dimension, nevertheless possesses intricate algebraic structure. This highlights that even within sets considered "negligible" from a measure-theoretic perspective, complex algebraic structures can still exist. Gaps in Our Intuition: The existence of such a subring challenges our intuitive understanding of the real line. We tend to visualize the real line as a continuous, "gapless" entity. However, this subring, despite being uncountable, occupies a "negligible" portion of the real line in terms of Hausdorff dimension. This suggests a counterintuitive picture where the real line, even after removing a "negligible" set, still retains its fundamental structure. In conclusion, the existence of an uncountable subring of ℝ with Hausdorff dimension zero underscores the remarkable richness and complexity of the real number line. It demonstrates that our typical notions of "size" based on cardinality or measure are insufficient to fully capture the intricate structure and diversity of subsets within the real numbers. It hints at a landscape far more nuanced and intricate than our intuition might initially suggest.

Core Concepts

This paper presents a method for constructing an uncountable subring of the real numbers (ℝ) that exhibits a Hausdorff dimension of zero, implying it also has Lebesgue measure zero.

Abstract

Bibliographic Information:

Baier, S., & Paul, S. (2024). An uncountable subring of R with Hausdorff dimension zero. arXiv:2411.13519v1 [math.NT].

Research Objective:

This paper aims to construct an uncountable subring of the real numbers (ℝ) that possesses a Hausdorff dimension of zero.

Methodology:

The authors define a specific set S consisting of zero and powers of 2. They then generate sumsets of S, denoted as nS, and construct sets An comprising real numbers formed by specific infinite series involving elements from nS. They prove that each An is an uncountable subgroup of ℝ and further demonstrate that each An has Hausdorff dimension zero by bounding the number of elements in nS less than a given value and analyzing the Hausdorff measure of specific coverings of An. Finally, they define the set A as the union of all An and prove that A is an uncountable subring of ℝ with Hausdorff dimension zero.

Key Findings:

For each natural number n, the set An constructed is an uncountable subgroup of ℝ with Hausdorff dimension zero.
The union of all An, denoted by A, forms an uncountable subring of ℝ, also exhibiting a Hausdorff dimension of zero.
The only rational numbers present within the constructed subring A are the dyadic rationals.

Main Conclusions:

The authors successfully construct an uncountable subring of the real numbers with Hausdorff dimension zero. This construction implies that the subring also has Lebesgue measure zero. Additionally, the authors establish that this subring is not a field as it only contains dyadic rationals.

Significance:

This work contributes to the understanding of the structure of the real numbers by demonstrating the existence of an uncountable subring with zero Hausdorff dimension. This result has implications for measure theory and fractal geometry.

Limitations and Future Research:

The paper focuses on a specific construction of a subring with desired properties. Exploring alternative constructions or generalizing this approach to other settings could be areas for future research. Additionally, investigating the properties of this specific subring further, such as its algebraic structure or topological characteristics, could provide deeper insights.

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Key Insights Distilled From

An uncountable subring of $\mathbb R$ with Hausdorff dimension zero

by Stephan Baie... at arxiv.org 11-21-2024

https://arxiv.org/pdf/2411.13519.pdf

$An uncountable subring of $\mathbb R$ with Hausdorff dimension zero$

Deeper Inquiries

Can this construction be generalized to create uncountable subrings with other specific Hausdorff dimensions between zero and one?

This is a very interesting question that the paper leaves open for further investigation.  While the provided construction specifically yields a subring with Hausdorff dimension zero, generalizing it to achieve other dimensions between zero and one is not immediately obvious. Here's why:

Controlling the Growth: The key to achieving Hausdorff dimension zero in the paper is the careful control of the growth rate of the set nS. The bound on gn(z) (the number of elements in nS less than z) ensures that the sets An,t can be covered efficiently by intervals of decreasing size. To achieve a specific Hausdorff dimension d between 0 and 1, one would need to construct sets analogous to nS with a growth rate carefully tuned to d. This would likely involve a deeper understanding of the interplay between additive number theory (how the sums in nS behave) and fractal geometry (how this growth rate translates to Hausdorff dimension).

Uncountability and Ring Structure:  It's not enough to simply control the Hausdorff dimension. The construction also needs to ensure the resulting set is uncountable and forms a ring.  Modifying the construction to target a specific Hausdorff dimension might inadvertently affect these properties. For instance, simply slowing down the growth rate of nS might lead to a countable set, and arbitrarily changing its structure might not result in closure under addition and multiplication.
In summary, while generalizing the construction to other Hausdorff dimensions is an intriguing possibility, it would require significant further research. It would demand a delicate balancing act between controlling the set's growth rate to achieve the desired dimension while preserving its uncountability and ring structure.

Could there be an uncountable subring of ℝ with Hausdorff dimension zero that is also a field?

It's highly unlikely. Here's why:

Fields and Inverses: The crucial difference between a ring and a field is the existence of multiplicative inverses for all nonzero elements. In a field, if x is a nonzero element, there exists an element x-1 such that x * x-1 = 1.

Hausdorff Dimension and Density:  Sets with Hausdorff dimension zero are "thin" in the sense that they have Lebesgue measure zero.  However, a subfield of ℝ cannot be too thin. If a subfield of ℝ contains an element x arbitrarily close to zero, then it must also contain its inverse x-1, which would be arbitrarily large. This suggests that a subfield cannot be "sparsely" distributed within ℝ like a set of Hausdorff dimension zero.

Corollary 1 Implication: Corollary 1 in the paper demonstrates that the constructed ring A only contains dyadic rationals (numbers of the form a/2k).  A field would need to contain the multiplicative inverses of all its nonzero elements. However, the multiplicative inverses of most dyadic rationals are not themselves dyadic rationals. For example, the inverse of 1/3 is 3, which is not a dyadic rational.
Therefore, while not explicitly proven in the paper, it's highly improbable to have an uncountable subring of ℝ that is both a field and has Hausdorff dimension zero. The requirements of a field seem fundamentally at odds with the "thinness" implied by Hausdorff dimension zero.

How does the existence of such a subring impact our understanding of the "size" and complexity of the real number line?

The existence of such a subring provides a fascinating glimpse into the intricate structure of the real number line, challenging our intuitive notions of "size" and complexity:

Beyond Countability:  We typically differentiate the "size" of infinite sets using cardinality. The real numbers are uncountable, while the rationals are countable. However, Hausdorff dimension offers a finer measure of size within the realm of uncountable sets. The existence of an uncountable subring with Hausdorff dimension zero reveals that even within the uncountable real numbers, there exist sets of dramatically different "sizes," some far "thinner" than others.

Hidden Algebraic Structure: The real numbers are not just a set; they form a field with rich algebraic properties. The constructed subring, while "small" in the sense of Hausdorff dimension, nevertheless possesses intricate algebraic structure. This highlights that even within sets considered "negligible" from a measure-theoretic perspective, complex algebraic structures can still exist.

Gaps in Our Intuition: The existence of such a subring challenges our intuitive understanding of the real line. We tend to visualize the real line as a continuous, "gapless" entity. However, this subring, despite being uncountable, occupies a "negligible" portion of the real line in terms of Hausdorff dimension. This suggests a counterintuitive picture where the real line, even after removing a "negligible" set, still retains its fundamental structure.
In conclusion, the existence of an uncountable subring of ℝ with Hausdorff dimension zero underscores the remarkable richness and complexity of the real number line. It demonstrates that our typical notions of "size" based on cardinality or measure are insufficient to fully capture the intricate structure and diversity of subsets within the real numbers. It hints at a landscape far more nuanced and intricate than our intuition might initially suggest.