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Dirichlet improvability in $L_p$ -norms

This paper provides a complete characterization of $F$ -Dirichlet improvable numbers in terms of continued fraction patterns for $L_p$ -norms, resolving open questions by Kleinbock and Rao regarding the full Hausdorff dimension of specific set differences and determining the precise range of $p$ for which the number $e$ is Dirichlet improvable.

Original authors: Nikolay Moshchevitin, Nikita Shulga

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nikolay Moshchevitin, Nikita Shulga

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 guess a secret number, $\alpha$ , by using fractions like $1/2$ , $3/4$ , or $22/7$ . In the world of mathematics, this is called Diophantine approximation. The goal is to see how closely a fraction can "hug" the secret number without actually being it.

For over a century, mathematicians have known a rule (Dirichlet's Theorem) that says: "No matter what your secret number is, you can always find a fraction close enough to it." However, for some very stubborn numbers, you can't get too close. These are called Badly Approximable numbers. They are the "uncooperative" numbers of the math world.

This paper, written by Nikolay Moshchevitin and Nikita Shulga, explores a new way of measuring "closeness." Instead of using the standard ruler (the usual distance), they use different shapes of "rulers" called $L_p$ -norms.

The Different Rulers

Think of these norms as different ways to measure distance in a city:

The Taxi Norm ( $L_1$ ): You can only drive along city blocks (up, down, left, right). The distance is the sum of the steps.
The Straight-Line Norm ( $L_2$ ): You can fly diagonally. This is the standard Euclidean distance.
The "Max" Norm ( $L_\infty$ ): You only care about the longest single step you took, ignoring the others.

The authors ask: If we change the ruler, does the list of "uncooperative" numbers change?

The Main Discovery: A Secret Code

The paper's biggest breakthrough is finding a "decoder ring" for these numbers. They discovered that whether a number is "improvable" (easy to approximate) or "non-improvable" (hard to approximate) depends entirely on the pattern of numbers inside its continued fraction.

A continued fraction is like a recipe for a number, written as a list of integers: $[a_0; a_1, a_2, a_3, \dots]$ .

The Old Rule: For the standard ruler ( $L_\infty$ ), a number is stubborn if its recipe contains huge numbers that keep getting bigger forever.
The New Rule: The authors found that for other rulers ( $L_1$ , $L_2$ , etc.), the "stubbornness" depends on specific symmetrical patterns in the recipe.

Imagine the recipe is a song.

For the Taxi Norm ( $L_1$ ), the song is stubborn if it has a very specific, almost-symmetrical chorus that repeats with growing volume.
For the Straight-Line Norm ( $L_2$ ), the song is stubborn if it has a specific pattern where two parts of the song multiply to equal 3.
For the Max Norm ( $L_\infty$ ), it's just about having huge numbers.

The authors mapped out exactly which patterns make a number stubborn for every type of ruler.

The Surprising Results

Using this pattern-mapping, they solved several puzzles that were previously unsolved:

The "In-Between" Numbers:
They proved that there is a massive, infinite crowd of numbers that are "improvable" (easy to approximate) with one ruler but "non-improvable" (hard) with another.
- Analogy: Imagine a number that is easy to catch with a net made of squares (Taxi norm) but impossible to catch with a net made of circles (Straight-line norm). The authors showed that the set of these "shape-shifting" numbers is so vast it has "full dimension" (it's as big as the entire number line in a mathematical sense).
The Mystery of the Number $e$ :
They applied their new rules to the famous number $e$ (the base of natural logarithms, roughly 2.718).
- They found that $e$ is easy to approximate with the Taxi ruler ( $L_1$ ) and the Straight-line ruler ( $L_2$ ).
- However, $e$ becomes stubborn (hard to approximate) if you use a ruler with a specific "stretch" between 2 and a special constant called $p_0$ (approx 2.57), or if you use a ruler even more stretched than that.
- In simple terms: $e$ is a chameleon. It behaves differently depending on which "ruler" you use to measure it.
The "Critical" Constant ( $p_0$ ):
There is a magical number, $p_0 \approx 2.57$ , where the rules of the game change completely. Below this number, the patterns required to be "stubborn" are one thing; above it, they are something else. The authors identified this tipping point and described exactly how the rules flip.

Why This Matters (According to the Paper)

The paper doesn't talk about building bridges or curing diseases. Instead, it solves a deep theoretical puzzle about the structure of numbers.

It connects the abstract shape of a "ruler" to the specific sequence of digits in a number's recipe.
It proves that the difference between these different types of numbers is not just a tiny speck, but a massive, complex landscape.
It provides a complete "instruction manual" for determining if a number is stubborn just by looking at its continued fraction pattern.

In short, the authors took a complex problem about measuring numbers with different shapes and turned it into a game of pattern recognition, revealing that the universe of numbers is far more diverse and structured than previously thought.

Technical Summary: Dirichlet Improvability in $L^p$ -Norms

Problem Statement

The paper addresses the classification of Dirichlet improvable numbers with respect to general $L^p$ -norms on $\mathbb{R}^2$ . While the classical case of the supremum norm ( $L^\infty$ ) is well-understood—where a number is Dirichlet improvable if and only if it is not badly approximable (Theorem B)—the behavior of these sets for other norms ( $1 \le p < \infty$ ) remained largely open.

Specifically, the authors tackle three interconnected problems:

Classification (Problem I): Can one characterize $p$ -Dirichlet improvable numbers ( $DI_p$ ) in terms of the patterns found in their regular continued fraction expansions, analogous to the classical result for $L^\infty$ ?
Metric Size of Differences (Problem II): Does the set difference $DI_p \setminus BA$ (numbers that are $p$ -improvable but not badly approximable) have full Hausdorff dimension? This was an open question posed by Kleinbock and Rao for the Euclidean norm ( $p=2$ ).
Comparative Size (Problem III): What is the size of the difference between sets of improvable numbers for two different norms, specifically $DI_{p_1} \setminus DI_{p_2}$ ?

Methodology

The authors employ a combination of geometry of numbers, dynamical systems, and metrical theory of continued fractions.

Dani's Correspondence and Critical Lattices:
The study utilizes Dani's correspondence to translate Diophantine approximation problems into the dynamics of lattice trajectories in the space of unimodular lattices. A number $\alpha$ is $F$ -Dirichlet non-improvable if and only if the trajectory of the lattice $\Lambda_\alpha(t)$ accumulates on a critical lattice for the unit ball $B_F$ of the norm $F$ .
The authors rely heavily on the classification of critical lattices for $L^p$ -unit balls ( $B_p$ ), a result completed by Glazunov, Golovanov, and Malyshev [14]. This classification reveals a structural bifurcation at a critical constant $p_0 \approx 2.57$ .
Pattern Analysis in Continued Fractions:
By analyzing the limit behavior of lattice trajectories converging to critical lattices, the authors derive necessary and sufficient conditions for $p$ -Dirichlet non-improvability. These conditions are expressed as the occurrence of specific palindromic or symmetric patterns in the partial quotients of the continued fraction expansion of $\alpha$ .
- For $2 < p < p_0$ , non-improvability corresponds to patterns like $x, 1, 1, y$ or $x, 2, y$ with large $x, y$ .
- For $p \in (1, 2) \cup (p_0, \infty)$ , the patterns depend on the continued fraction expansion of a specific constant $\varsigma_p$ (the root of $\varsigma + (1+\varsigma)^p = 2$ ).
Hausdorff Dimension Bounds:
To prove that certain sets have full Hausdorff dimension, the authors construct subsets of badly approximable numbers ($BA$) by inserting specific patterns into their continued fractions. They utilize a symbolic space mapping technique (Lemma 9.1) to show that inserting these patterns preserves the full dimension of the underlying set while altering the Diophantine properties. This involves proving that the insertion map is locally Lipschitz and its inverse is locally Hölder continuous.

Key Contributions and Results

1. Complete Classification (Theorem 3.1)

The paper provides a complete and effective characterization of the set $DI_p^c$ (the complement of $p$ -Dirichlet improvable numbers) for all $p \in [1, \infty)$ .

Case $2 < p < p_0$ : $\alpha \in DI_p^c$ iff its continued fraction contains patterns $x, 1, 1, y$ or $x, 2, y$ with $\min(x, y) \to \infty$ .
Case $p \in (1, 2) \cup (p_0, \infty)$ : The condition depends on $\varsigma_p$ . If $\varsigma_p$ is rational, specific finite palindromic patterns involving the digits of $\varsigma_p$ must occur. If $\varsigma_p$ is irrational, arbitrarily long palindromic patterns derived from the continued fraction of $\varsigma_p$ must occur.
Case $p=1$ and $p=2$ : The characterization involves more complex symmetric patterns and, for $p=2$ , pairs of numbers $\beta, \beta^*$ satisfying $\beta \cdot \beta^* = 3$ .

2. Resolution of Open Questions on Dimension

Theorem 1.1: For any $p \in [1, \infty)$ , the set $DI_p \setminus BA$ has full Hausdorff dimension (dimension 1). This resolves the open question by Kleinbock and Rao for $p=2$ and generalizes it to all $L^p$ norms. The authors explicitly note that this fails for $p=\infty$ (where $DI_\infty \setminus BA = \mathbb{Q}$ ).
Theorem 1.2: The sets $DI_2 \setminus DI_1$ and $DI_1 \setminus DI_2$ both have full Hausdorff dimension. This demonstrates that the sets of improvable numbers for different $L^p$ norms are distinct and that their differences are "large" in a fractal sense.

3. Characterization of Specific Constants

Corollary 3.9: The authors apply their classification to the number $e$ . They prove that $e \in DI_p$ if and only if $p \in (1, 2) \cup (p_0, \infty)$ . Consequently, $e$ is not $p$ -Dirichlet improvable for $p \in [2, p_0]$ .
Corollary 3.8: The set of values $p$ for which there exist $p$ -Dirichlet non-improvable badly approximable numbers is dense in $(1, 2) \cup (p_0, \infty)$ , has zero Lebesgue measure, and has full Hausdorff dimension.

4. Verification of Minkowski's Conjecture

The paper includes an independent verification of the proof of Minkowski's conjecture regarding critical lattices for $L^p$ balls (Appendix A), addressing the lack of code in the original proof by Glazunov et al. [14].

Significance and Claims

The paper claims to provide the first complete and effective classification of Dirichlet improvable numbers for $L^p$ -norms in terms of continued fraction patterns. This bridges the gap between the classical theory of $L^\infty$ and the more complex geometry of other norms.

The significance lies in:

Unification: It unifies the understanding of Dirichlet improvability across the entire family of $L^p$ norms, revealing a phase transition at $p_0 \approx 2.57$ .
Metric Resolution: It settles the question of the "size" of the sets $DI_p \setminus BA$ and $DI_{p_1} \setminus DI_{p_2}$ , showing they are not merely non-empty but have the maximal possible Hausdorff dimension.
Specific Applications: It provides a definitive answer regarding the Diophantine properties of the number $e$ under $L^p$ norms, a specific instance of the broader classification.

The authors emphasize that their results are derived from the classification of critical lattices and the theory of continued fractions, avoiding heuristic arguments. The work confirms that while almost all numbers are non-improvable (Theorem C), the set of improvable numbers is structurally rich and varies significantly with the choice of norm $p$ .