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The Dirichlet spectrum with respect to $L_1$ norm is $\left[\frac12,1\right]$

This paper proves that the one-dimensional Dirichlet spectrum with respect to the $L_1$ norm is the interval $[\frac{1}{2}, 1]$ , which is equivalent to the Minkowski spectrum being $[\frac{1}{4}, \frac{1}{2}]$ , and further establishes that the Hausdorff dimension of the level sets of the Minkowski constant is strictly greater than $\frac{1}{2}$ for values in $(\frac{1}{4}, \frac{1}{2}]$ while equaling $\frac{1}{2}$ at the boundary.

Original authors: Nikita Shulga

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: 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 (an irrational number like $\pi$ or $\sqrt{2}$ ) using simple fractions (like $22/7$ or $355/113$ ). Mathematicians have long been fascinated by how "well" you can guess these numbers.

This paper is about a specific game of guessing, but with a twist in the rules.

The Game: Guessing with a New Ruler

Usually, when mathematicians measure how close a guess is, they use a standard "ruler" (the Euclidean distance, like measuring the straight line between two points). This paper changes the ruler. Instead of a straight line, they use an L1 ruler, which is like measuring distance by walking only along city blocks (Manhattan distance). You can't cut diagonally; you have to go up/down and left/right.

The question the author asks is: What are the possible "best scores" you can get in this city-block guessing game?

In math terms, this set of all possible best scores is called the Dirichlet Spectrum.

The Big Discovery: A Perfect Interval

For a long time, mathematicians knew the spectrum existed, but they didn't know exactly what shape it took. Was it a scattered collection of dots? A weird, broken line?

The main result of this paper is simple and beautiful:
The set of all possible best scores for this specific "city-block" game is a perfect, unbroken interval from 0.5 to 1.

Think of it like a volume knob. If you turn the knob, you can get any volume level between 0.5 and 1. You can't get 0.4, and you can't get 1.1, but you can get 0.5001, 0.9999, or anything in between. This is the first time a one-dimensional version of this problem has been fully solved with such a clean answer.

The Secret Code: The "Minkowski" Connection

To solve this, the author didn't just look at the guessing game directly. They realized the game was secretly connected to a special code called the Minkowski Diagonal Continued Fraction.

Imagine a number written as a long string of digits (a continued fraction). Most numbers have random-looking digits. But this special code filters the digits, keeping only the ones that follow a very specific rule (related to the "city-block" distance).

The author proved that the "best score" in the guessing game is directly linked to the behavior of this special code. Specifically:

The "Minkowski Spectrum" (the scores of this special code) is the interval 0.25 to 0.5.
The "Dirichlet Spectrum" (the scores of the original guessing game) is exactly twice that: 0.5 to 1.

It's like finding out that two different languages are actually speaking the same story, just with different volume settings.

The "Level Sets": How Many Numbers Have a Specific Score?

The paper also asks a deeper question: If I want a number that gets a specific score (say, exactly 0.6), how many such numbers are there?

In math, we measure "how many" using something called Hausdorff dimension.

A single point has dimension 0.
A line has dimension 1.
A surface has dimension 2.

The author found that for almost any score between 0.25 and 0.5 (in the Minkowski world), the set of numbers that achieve that score is surprisingly "large."

They proved that the "size" (dimension) of these groups of numbers is strictly greater than 0.5.
This means these groups are "fatter" than a simple line, even though they are still "thinner" than a full 2D surface.

However, there is a special case at the very bottom of the scale (the score 0.25). Here, the author proved the size is exactly 0.5. It's a sharp boundary where the "fatness" of the group changes.

The Magic Trick: Building the Numbers

How did the author prove these numbers exist? They used a clever construction method, like building a tower with blocks.

They took a target score they wanted to achieve.
They used a special algorithm to break a number into two parts.
They built a giant, infinite string of digits by alternating between these two parts, separated by huge "separators" (very large numbers).
This specific pattern forces the "score" of the number to settle exactly on the target value.

Summary

In plain English, this paper says:

If you play a specific number-guessing game using "city-block" distance, your best possible scores fill up a perfect, unbroken range from 0.5 to 1.
This range is the largest possible range allowed by the laws of mathematics for this type of game.
For almost every score in that range, there is a surprisingly large (though not infinite) collection of numbers that hit that score exactly.
The author figured out exactly how to build these numbers using a special pattern of digits.

This is a foundational result in number theory, mapping out the landscape of how well we can approximate numbers under these specific rules.

Technical Summary: The Dirichlet Spectrum with Respect to the $L^1$ Norm

Problem Statement
The paper investigates the Dirichlet spectrum $D[F]$ associated with a norm $F$ on $\mathbb{R}^2$ . For an irrational number $\alpha$ , the Dirichlet constant $d_F(\alpha)$ is defined via the first successive minimum of the lattice $\Lambda_\alpha(t) = \text{diag}(t^{-1}, t) \begin{pmatrix} 1 & 0 \\ -\alpha & 1 \end{pmatrix} \mathbb{Z}^2$ with respect to the unit ball of $F$ . The normalized Dirichlet constant is given by $\delta_F(\alpha) = \Delta_F \cdot (d_F(\alpha))^2$ , where $\Delta_F$ is the critical determinant of the unit ball. The spectrum $D[F]$ is the set of all such values $\delta_F(\alpha)$ for $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ .

While the structure of the classical Dirichlet spectrum (associated with the sup-norm, $L^\infty$ ) is known to be complex, containing both discrete points and a continuous "Hall's ray," the structure for other norms remains less understood. This paper focuses specifically on the $L^1$ norm, defined by $F^{[1]}(x, y) = |x| + |y|$ . The central problem is to determine the exact set $D[1] = \{ \delta_{F^{[1]}}(\alpha) : \alpha \in \mathbb{R} \setminus \mathbb{Q} \}$ .

Methodology
The author employs a multi-step strategy connecting Diophantine approximation, continued fractions, and fractal geometry:

Connection to Minkowski Spectrum: The paper establishes a direct link between the $L^1$ Dirichlet spectrum and the Minkowski spectrum $M$ , associated with the Minkowski diagonal continued fraction (DCF). It is shown that $\delta_{F^{[1]}}(\alpha) = 2m(\alpha)$ , where $m(\alpha)$ is the Minkowski constant defined via the limsup of $t \mu_\alpha(t)$ , with $\mu_\alpha(t)$ being the linear interpolation of the lower convex hull of the set $\{(q, \|q\alpha\|)\}$ . Consequently, $D[1] = 2M$ . The problem is thus reduced to characterizing the Minkowski spectrum $M$ .
Functional Representation: Utilizing Moshchevitin's results, the Minkowski constant $m(\alpha)$ is expressed as the limsup of values derived from two functions, $G(x,y)$ and $F(x,y)$ , which depend on the tails of the regular continued fraction expansion of $\alpha$ . The function $F$ corresponds to consecutive convergents satisfying Legendre's inequality, while $G$ corresponds to "skipped" convergents. To facilitate analysis, the author introduces a change of variables transforming $F$ into a function $K(u,v)$ , defined on positive real numbers, which simplifies the estimation of bounds.
Sum Decomposition Algorithm: A crucial tool is a recently discovered algorithm (from [27] and [4]) that decomposes any irrational $s \in (0,1)$ into a sum of two continued fractions, $s = \beta + \gamma$ , where the partial quotients of $\beta$ and $\gamma$ tend to infinity. This decomposition allows for the explicit construction of numbers with specific properties.
Explicit Construction of $\alpha$ : To prove that a specific value $m \in (1/4, 1/2)$ is in the spectrum, the author constructs an irrational $\alpha$ by concatenating blocks derived from the decomposition of a target $s = 4m-1$ . The construction involves:
- Creating blocks $W_N = (b_N, \dots, b_1, 1, c_1, \dots, c_N)$ based on the partial quotients of $\beta$ and $\gamma$ .
- Inserting these blocks with large "separator" partial quotients $d_j$ to isolate the behavior of the central digit '1'.
- Analyzing the resulting continued fraction to show that the limsup of the Minkowski constant is exactly determined by the central "skipped" terms (associated with the digit 1), while all other cuts yield strictly smaller values.
Hausdorff Dimension Analysis: For the level sets $\Theta_m = \{ \alpha : m(\alpha) = m \}$ , the paper uses the constructed blocks to embed sets of numbers with restricted partial quotients. By applying distortion arguments and properties of continued fraction cylinders, the author estimates the Hausdorff dimension of these sets.

Key Results

Theorem 1.1 (Main Result): The Dirichlet spectrum for the $L^1$ norm is the closed interval:
$D[1] = \left[ \frac{1}{2}, 1 \right].$
Equivalently, the Minkowski spectrum is $M = [1/4, 1/2]$ . This result confirms that for the $L^1$ norm, the spectrum occupies the entire theoretically possible range for symmetric norms (since $1/2 \le \delta_F(\alpha) \le 1$ generally).
Theorem 1.3 (Dimension of Level Sets): For any $m \in (1/4, 1/2)$ , the level set $\Theta_m$ has a Hausdorff dimension strictly greater than $1/2$ . Specifically, if $s = 4m-1$ and $A$ is sufficiently large, $\dim_H \Theta_m \ge \delta_A$ , where $\delta_A$ is the dimension of the set of numbers with partial quotients $\ge A$ .
Theorem 7.3: For any $m \in (1/4, 1/2)$ , $\dim_H \Theta_m < 1$ . This distinguishes these level sets from the generic case ( $m=1/2$ ), which has full Lebesgue measure.
Theorem 7.6: The dimension of the endpoint level set is exactly $\dim_H \Theta_{1/4} = 1/2$ .
Corollary 7.7: The limit of the dimensions as $m$ approaches $1/4$ from above is $1/2$ .

Significance and Claims
The paper claims to provide the first fully described structure of a one-dimensional Diophantine spectrum. While multi-dimensional Dirichlet spectra have recently been shown to be intervals, the one-dimensional case is typically more complex. By proving that $D[1]$ is a closed interval, the work resolves the structure for the $L^1$ norm.

Furthermore, the paper demonstrates that the level sets of the Minkowski constant are "large" in a fractal sense (dimension $> 1/2$ ) for all interior points of the spectrum, while the endpoints have distinct dimensional properties. The methodology relies on the explicit construction of numbers via continued fraction concatenation, bridging the gap between the algebraic properties of the Minkowski constant and the geometric properties of the associated sets.

The author notes that the result relies on the specific properties of the $L^1$ norm and the Minkowski diagonal continued fraction, and does not claim to generalize these structural results to arbitrary norms without further investigation. The work is presented as a solution to a problem posed by Moshchevitin regarding the Minkowski spectrum.