On The Hausdorff Dimension Of Two Dimensional Badly Approximable Vector

This paper establishes that for any ball in the unit square, the Hausdorff dimension of the intersection with the set of weighted badly approximable vectors equals that of the corresponding set of weighted approximable vectors, providing an explicit formula for this dimension in terms of the weight parameters.

The Gist

Imagine you are standing in a vast, empty square room (the unit square $[0, 1]^2$). Scattered throughout this room are thousands of invisible "target zones" centered around rational numbers (fractions like $1/2, 3/7, 22/7$, etc.).

In the world of mathematics, there is a game called Diophantine Approximation. The goal is to see how closely you can get to these rational targets using a specific set of rules.

The Game: "How Close Can You Get?"

Imagine the rational numbers are like lighthouses.

• The Rule: You are allowed to stand near a lighthouse if you are within a certain distance.
• The Twist: This distance isn't the same for everyone. It depends on the "size" of the fraction (its denominator).
  • Small fractions (like $1/2$) have large target zones.
  • Huge fractions (like $1/1,000,000$) have tiny, needle-thin target zones.

This paper studies a specific version of this game where the target zones shrink at different rates for the horizontal and vertical directions. We call this the "Weighted" setting.

The Two Types of Players

The paper looks at two groups of people standing in the room:

1. The "Good Approximators" ($A_m$): These are people who can get infinitely close to the lighthouses. No matter how small the target zone gets, they can always find a spot inside it. They are the "winners" of the approximation game.
2. The "Badly Approximable" Vectors ($B_m$): These are the tricky ones. They are also good approximators (they get close infinitely often), BUT they have a safety margin. They can never get too close. If you try to shrink the target zone by even a tiny bit (multiplying the distance by a small number $c$), they suddenly fall out of the zone. They are "badly approximable" because they refuse to get arbitrarily close to the perfect rational points.

The Big Question: How "big" is the set of these "Badly Approximable" people?

In math, "size" can mean area (Lebesgue measure) or "fractal dimension" (Hausdorff dimension).

• If you look at the area, the "Badly Approximable" set is actually zero. It's like a dust cloud; it has no volume.
• But, it's not empty. It's a fractal. The paper asks: What is the fractal dimension of this dust cloud?

The Author's Discovery

Yi Lou, the author, solves this puzzle for a 2-dimensional room with specific rules (where the "weights" or shrinking rates are different for the x and y axes).

He proves that the "size" (dimension) of this dust cloud is exactly the same as the size of the set of all "Good Approximators."

The Formula:
The dimension is determined by a "tug-of-war" between two numbers. The final size is the smaller of these two values:

1. A value based on the "slower" shrinking rate.
2. A value based on the "faster" shrinking rate.

Think of it like a bucket with a hole. The water level (the dimension) is determined by the size of the biggest hole, not the sum of all holes.

How Did He Prove It? (The Construction Analogy)

To prove this, Lou had to build a specific "fractal dust cloud" and show it fits the rules. He used a method called a Cantor Construction, which is like a game of "Keep the Room, Remove the Targets."

Here is the step-by-step analogy:

1. The Grid: Imagine the room is tiled with millions of tiny tiles.
2. The Cleanup: He looks at the rational lighthouses. For every lighthouse, he draws a "danger zone" around it.
3. The Removal: He sweeps away any tile that touches a danger zone.
   • Crucial Detail: He doesn't just remove the exact spot; he removes a slightly larger buffer zone to ensure no one can get too close.
4. The Survivors: After doing this infinitely many times (removing zones for larger and larger fractions), the tiles that remain form his "Badly Approximable" set.

The Challenge:
If you remove too much, you might wipe out the whole room (leaving nothing). If you remove too little, the survivors might still be able to get too close to the lighthouses.

The Solution:
Lou used a clever "Mass Distribution" technique. Imagine pouring water (mass) into the room.

• He carefully calculates how much water to pour into the remaining tiles at each step.
• He proves that even after removing all the danger zones, there is still enough "water" left in the remaining dust cloud to prove it has a specific, non-zero fractal dimension.
• He shows that the "water" spreads out just enough to fill the space defined by his formula, but not so much that it spills over into the forbidden "too close" zones.

Why Does This Matter?

This paper is a piece of a larger puzzle in number theory.

• Unweighted vs. Weighted: Previous mathematicians solved this for the "unweighted" case (where both directions shrink at the same rate). Lou extended this to the "weighted" case (where one direction shrinks faster than the other).
• Independence: Interestingly, while other mathematicians recently found a different way to solve a similar problem, Lou's proof is unique. He didn't use their shortcuts; he built the fractal from the ground up using geometry and careful counting.

The Takeaway

In simple terms: Even though the "Badly Approximable" numbers are rare (they have zero area), they are surprisingly "thick" in a fractal sense.

Yi Lou showed us exactly how thick they are in a 2D world where the rules of closeness are different for the horizontal and vertical directions. He proved that the "thickness" is determined by the most restrictive rule in the game, and he built a mathematical model to prove it beyond any doubt.

Technical Summary

Here is a detailed technical summary of the paper "On the Hausdorff Dimension of Two Dimensional Weighted Badly Approximable Vectors" by Yi Lou.

1. Problem Statement and Context

The paper investigates the Hausdorff dimension of the set of weighted badly approximable vectors in two dimensions.

• Definitions:

  • Let $\Psi = (\psi_1, \dots, \psi_m)$ be an approximation function. The set of $\Psi$-approximable vectors, $A_m(\Psi)$, consists of vectors $x \in [0,1]^m$ such that $|qx_i - p_i| < \psi_i(q)$ for infinitely many integers $q$ and $p$.
  • The set of weighted $\Psi$-badly approximable vectors, $B_m(\Psi)$, is defined as $A_m(\Psi) \setminus \bigcup_{0<c<1} A_m(c\Psi)$. Essentially, these are vectors that are approximable at the rate $\Psi$ but not significantly better (by any constant factor).
  • The paper focuses on the power-law setting where $\psi_i(q) = q^{-\tau_i}$ for a weight vector $\tau = (\tau_1, \dots, \tau_m)$. The set is denoted $B_m(\Psi_\tau)$.

• Known Results:

  • If $\sum \tau_i < 1$, Dirichlet's Theorem implies $B_m(\Psi_\tau)$ is empty.
  • If $\sum \tau_i = 1$, Schmidt proved $\dim_H B_m(\Psi_\tau) = m$.
  • If $\sum \tau_i > 1$, Wang and Wu computed $\dim_H A_m(\Psi_\tau)$. For the unweighted case ($\tau_1 = \dots = \tau_m$), Koivusalo et al. proved that the local dimension of $B_m(\Psi_\tau)$ equals that of $A_m(\Psi_\tau)$.
  • The Gap: While recent independent work by Bandi and Fregoli established that $\dim_H B_m(\Psi_\tau) = \dim_H A_m(\Psi_\tau)$ for general weights, their proof relies on different methods. This paper aims to provide an independent proof specifically for the 2-dimensional weighted case ($m=2$) using geometric constructions, thereby elucidating the local structure of these sets.

2. Main Result

Theorem 1.2: Let $\tau = (\tau_1, \tau_2)$ with $\tau_1 \ge \tau_2 > 0$ and $\tau_1 + \tau_2 > 1$. For any ball $B \subseteq [0, 1]^2$, the local Hausdorff dimension is given by:
$$\dim_H(B \cap B_2(\Psi_\tau)) = \dim_H A_2(\Psi_\tau) = \min \left\{ \frac{3 + \tau_1 - \tau_2}{1 + \tau_1}, \frac{3}{1 + \tau_2} \right\}$$
This result confirms that the "badly approximable" subset retains the full dimension of the "approximable" set within any local region.

3. Methodology

The proof employs a Cantor-type construction combined with the Mass Distribution Principle. The strategy involves constructing a specific Cantor subset $D_\epsilon(B) \subseteq B \cap B_2(\Psi_\tau)$ and showing its dimension is arbitrarily close to the target value $s$.

The proof is divided into four main steps:

Step 1: Construction of the Set $C_\tau(N)$

The authors construct a set $C_\tau(N)$ designed to avoid neighborhoods of rational points, ensuring elements are "badly approximable" relative to a slightly modified function.

• Simplex Lemma: Used to show that rational points with bounded denominators within small rectangles lie on specific hyperplanes (lines in 2D).
• Iterative Removal: The unit square is partitioned into a hierarchy of rectangles. At each level $n$, neighborhoods of hyperplanes containing rational points with denominators in a specific range ($N^{n-1} \le q < N^n$) are removed.
• Result: $C_\tau(N)$ is the intersection of these nested sets. It is proven that $C_\tau(N)$ avoids neighborhoods of all rational points $p/q$ with a specific radius depending on $q$ and $\tau$.
• Measure: It is shown that for large $N$, the Lebesgue measure of $C_\tau(N)$ is close to 1, meaning very little mass is removed locally.

Step 2: The Set of Leading Rationals

A subset of rational points, denoted $Q(N, \tau)$, is defined as "leading rationals."

• These are rationals $p/q$ that lie on the hyperplanes removed during the construction of $C_\tau(N)$ but are "active" in the sense that their neighborhoods intersect the surviving rectangles.
• Key Structural Lemmas:
  • Every point in $C_\tau(N)$ is approximable by infinitely many leading rationals.
  • For any leading rational $p/q$, there exists a surviving rectangle $R$ in the construction such that $R$ is contained within a specific neighborhood of $p/q$ and has a volume comparable to that neighborhood.

Step 3: Construction of the Cantor Subset $D_\epsilon(B)$

A Cantor set $D_\epsilon(B)$ is constructed inside a ball $B$ using the leading rationals.

• TG,R Lemma: A Vitali-type covering lemma is adapted for rectangles. It ensures that within any surviving rectangle $R$, one can find a disjoint collection of smaller rectangles (associated with leading rationals) that cover a significant proportion of the measure of $R$.
• Shrinking: The construction involves "shrinking" the covering rectangles to ensure the resulting limit set lies within $A_2(\Psi_\tau)$ (approximable) while the removal process in Step 1 ensures it lies within $B_2(\Psi_\tau)$ (badly approximable).
• Result: $D_\epsilon(B)$ is a subset of $B \cap B_2(\Psi_\tau)$.

Step 4: Mass Distribution and Dimension Estimation

The Mass Distribution Principle is applied to $D_\epsilon(B)$.

• Measure Construction: A probability measure $\mu$ is defined inductively on the layers of the Cantor set. The mass is distributed proportionally to the Lebesgue measure of the rectangles at each step.
• Parameter Tuning: The indices $G_n$ (which control the density of rationals selected at each step) are chosen carefully to satisfy specific growth conditions.
• Verification: The authors verify that for any ball $F$ of radius $r$, the measure satisfies $\mu(F) \le C r^{s-\epsilon}$.
  • The proof involves a case analysis based on the radius $r$ relative to the denominators of the rational centers.
  • It utilizes the specific geometry of the weighted rectangles (side lengths $q^{-1-\tau_i}$) and the covering properties derived in Step 3.
• Conclusion: By the Mass Distribution Principle, $\dim_H D_\epsilon(B) \ge s - \epsilon$. Since $\epsilon$ is arbitrary, the dimension is at least $s$. The upper bound is trivial as $B \cap B_2(\Psi_\tau) \subseteq A_2(\Psi_\tau)$.

4. Key Contributions

1. Independent Proof: Provides a self-contained, geometric proof for the dimension of weighted badly approximable vectors in dimension 2, independent of the recent measure-theoretic results by Bandi and Fregoli.
2. Extension to Weighted Case: Successfully extends the unweighted results of Koivusalo et al. to the weighted setting ($\tau_1 \neq \tau_2$), handling the asymmetry in the approximation rates.
3. Local Dimension Formula: Establishes a sharp formula for the local Hausdorff dimension, confirming that the dimension is determined by the "worst" direction of approximation (the minimum of the two derived expressions).
4. Technical Innovation: Introduces a refined TG,R Lemma (a variation of the Vitali covering lemma for rectangles) and a specific iterative removal process based on the Simplex Lemma to handle the weighted constraints.

5. Significance

• Diophantine Approximation: This work deepens the understanding of the fractal geometry of sets of badly approximable numbers, a central topic in metric number theory.
• Methodological Value: The geometric construction techniques (Cantor sets, mass distribution on weighted rectangles) offer a template for analyzing similar problems in higher dimensions or with different approximation functions.
• Completeness: It fills a specific gap in the literature regarding the 2-dimensional weighted case, providing a rigorous foundation that complements broader, more abstract results.

In summary, Yi Lou's paper rigorously demonstrates that even when restricting to vectors that are "badly" approximable (i.e., not approximable better than a constant factor), the set remains large in the sense of Hausdorff dimension, matching the dimension of the set of all vectors approximable at that rate.