Pólya's conjecture up to $\epsilon$-loss and quantitative estimates for the remainder of Weyl's law

This paper establishes an $\epsilon$-loss version of Pólya's conjecture for bounded Lipschitz domains by providing explicit quantitative estimates for the Weyl law remainder without relying on Neumann eigenvalues, thereby reducing the conjecture to a computational problem and identifying broader classes of domains, including irregular shapes and strip-tiling domains, that satisfy the conjecture or even exhibit stronger eigenvalue bounds.

The Gist

Imagine you have a mysterious, irregularly shaped room (let's call it Ω). You want to know how many distinct musical notes (or "vibrations") this room can produce if you strike its walls. In mathematics, these notes are called Dirichlet eigenvalues, and they are numbered $1, 2, 3, \dots$ from the lowest pitch to the highest.

For over a century, mathematicians have been trying to predict exactly how many notes exist below a certain pitch. This is known as Weyl's Law. It's like having a rough map that tells you, "If you go up to pitch $X$, you'll find roughly $Y$ notes." The map is based on the volume (size) of the room.

However, the map isn't perfect. There's always a "remainder" or an error term. The big question, posed by the famous mathematician George Pólya in 1954, was: Is the actual number of notes always less than or equal to the number predicted by the volume map?

Pólya proved this is true for rooms that can tile a floor perfectly (like squares or hexagons), but for weird, jagged, or irregular rooms, it remained an unsolved mystery.

The Main Breakthrough: "The $\epsilon$-Loss"

This paper by Renjin Jiang and Fanghua Lin doesn't solve the mystery for every single note in every room immediately. Instead, they found a clever workaround.

Think of it like this: Pólya's original guess was that the room can hold exactly $N$ notes. The authors say, "Okay, let's be slightly generous. Let's say the room can hold $N \times (1 + \epsilon)$ notes, where $\epsilon$ is a tiny, tiny bit of extra space (like 1% or 0.1%)."

They proved that for any room with a reasonably well-behaved boundary (a "Lipschitz domain"), if you look at the high-pitched notes (the ones with very high energy), the number of notes is indeed less than this slightly inflated prediction.

The "Computational" Twist:
The paper shows that to prove Pólya's strict conjecture for a specific room, you only need to check the notes up to a certain "cutoff" pitch. Once you pass that pitch, the math guarantees the rule holds. This turns a massive, impossible theoretical problem into a manageable computer calculation problem. You just need to crunch the numbers for the lower notes, and the high notes take care of themselves.

The "Strip-Tiling" Secret

The authors discovered a special class of shapes they call "Strip-Tiling Domains."

Imagine a long hallway. If you can take your weirdly shaped room, rotate it, and slide it along the hallway to cover the entire floor without gaps or overlaps, it's a strip-tiling domain.

• The Surprise: For these shapes, the room is actually more efficient than Pólya originally guessed. It holds fewer notes than the volume map predicts.
• The Triangle Example: This is huge for triangles! Since any triangle can tile a plane (you can fit them together perfectly), the authors prove that Pólya's conjecture is true for every single triangle, and in fact, the estimate is even better than expected.

The "Swiss Cheese" Strategy

What if you have a perfect shape (like a big square) and you punch holes in it (removing cubes)? Does the rule still hold?

The authors show that if you start with a shape that follows the rule (like a tiling shape or a triangle) and you remove a collection of small cubes (like taking bites out of a cookie), the rule still holds, provided the "surface area" of the original shape is large enough compared to the total size of the holes.

They call this the "Admissible Class" of cubes. It's like saying, "As long as the cookie isn't too full of holes, the rule about the number of notes remains valid."

The "Whitney Decomposition" (The Math Tool)

To prove these results, the authors used a technique called Whitney Decomposition.

• The Analogy: Imagine you have a jagged, irregular shape. To understand it, you don't look at the whole mess at once. Instead, you cover it with a grid of tiny, non-overlapping squares (like a mosaic).
• The Magic: They used this grid to count the notes in the tiny squares and then added them up. By carefully managing the "error" at the edges of these squares, they could create a precise "upper bound" (a ceiling) for the number of notes. This allowed them to prove that the number of notes never exceeds the limit, even with the messy boundaries.

Summary of What They Claim

1. $\epsilon$-Loss Version: For any bounded room, if you look at high enough notes, the count is strictly less than $(1 + \epsilon)$ times the volume prediction. This reduces the problem to a computer check for lower notes.
2. Better than Expected: For "Strip-Tiling" shapes (including all triangles), the number of notes is actually lower than the standard prediction, not just lower than the loose prediction.
3. Holes are Okay: You can remove a specific type of "Swiss cheese" pattern (cubes) from a valid shape, and the rule still holds, as long as the original shape was big enough relative to the holes.
4. No "Neumann" Tricks: Previous methods often relied on comparing the room to a "Neumann" version (a room with different boundary rules). The authors found a new way to prove this using only the "Dirichlet" rules (the standard vibrating walls), making their proof cleaner and more direct.

In short, the paper says: "We can't prove the rule for every single note in every single weird shape yet, but we can prove it for the high notes, and we've shown that for many specific shapes (like triangles and tiled strips), the rule is actually even stronger than we thought."

Technical Summary

Technical Summary: Pólya's Conjecture up to $\epsilon$-Loss and Quantitative Estimates for the Remainder of Weyl's Law

Problem Statement
The paper addresses two interconnected problems in spectral geometry regarding the Dirichlet Laplacian on bounded domains $\Omega \subset \mathbb{R}^n$ ($n \geq 2$):

1. Pólya's Conjecture: The conjecture states that for any bounded open domain $\Omega$, the $k$-th Dirichlet eigenvalue $\lambda_k(\Omega)$ satisfies the inequality $k \leq \frac{|\Omega|\omega(n)}{(2\pi)^n} \lambda_k(\Omega)^{n/2}$ for all $k \in \mathbb{N}$. While proven for tiling domains and specific shapes (balls, sectors, annuli), it remains open for general domains.
2. Quantitative Remainder of Weyl's Law: Weyl's law provides the asymptotic behavior of the eigenvalue counting function $N_\Omega^D(\lambda) = \#\{k : \lambda_k(\Omega) < \lambda\}$. The remainder $R_\Omega(\lambda) = N_\Omega^D(\lambda) - \frac{|\Omega|\omega(n)}{(2\pi)^n} \lambda^{n/2}$ is known to be $o(\lambda^{n/2})$, but previous results often lacked explicit constants or uniform bounds valid for all eigenvalues, particularly on rough (Lipschitz) domains.

Methodology
The authors employ a combination of refined eigenvalue estimates, geometric decomposition, and monotonicity arguments:

• Refined Estimates on Product and Strip-Tiling Domains: Building on Laptev's work on product domains and Pólya's tiling method, the authors derive sharper upper bounds for eigenvalue counting functions on product domains $\Omega_1 \times \Omega_2$ and "strip-tiling" domains (domains that can tile a strip $R^{n-1} \times (0, w)$ via isometries in the first $n-1$ directions). These estimates include explicit negative lower-order terms, improving upon the standard Pólya bound.
• Whitney Decomposition: To handle general Lipschitz domains, the authors utilize a Whitney decomposition of the domain and its complement. This allows them to bound the counting function by summing contributions from dyadic cubes, avoiding the use of Neumann eigenvalues which are central to Courant's classical Dirichlet-Neumann bracketing method.
• Quantitative Lower Bounds on Cubes: The paper establishes explicit lower bounds for the counting function on cubes (and by extension, cuboids) by analyzing the geometry of lattice points intersecting balls. These bounds are crucial for estimating the "error" introduced when removing sub-domains (like cubes) from a larger domain.
• Monotonicity and Domain Perturbation: The strategy involves enclosing a general domain $\Omega$ within a larger "nice" domain $T$ (such as a strip-tiling domain) and analyzing the remainder $T \setminus \Omega$. By proving that $T$ satisfies better-than-Pólya estimates and that the removed part $T \setminus \Omega$ has a controlled eigenvalue count, the authors derive bounds for $\Omega$.

Key Contributions and Results

1. Uniform Quantitative Remainder Estimate (Theorem 1.5):
   The paper provides a new proof of a uniform estimate for the remainder of Weyl's law on bounded Lipschitz domains with explicit constants. For any $k \in \mathbb{N}$:
   $$k \leq \frac{|\Omega|\omega(n)}{(2\pi)^n} \lambda_k(\Omega)^{n/2} + C(n, \Omega, \lambda_k) \frac{\omega(n)}{(2\pi)^n} \lambda_k(\Omega)^{\frac{n-1}{2}}$$
   where the constant $C$ depends explicitly on the Lipschitz constant $C_{\text{Lip}}(\Omega)$, the boundary area, and the geometry of the minimal admissible rectangle containing $\Omega$. A corresponding lower bound is also established. This result is derived without relying on Neumann eigenvalues.

2. $\epsilon$-Loss Version of Pólya's Conjecture (Theorem 1.3):
   The authors prove that for any $\epsilon \in (0, 1)$, there exists an explicitly computable threshold $\Lambda(\epsilon, \Omega)$ such that for all eigenvalues $\lambda_k(\Omega) \geq \Lambda(\epsilon, \Omega)$, the $\epsilon$-loss version of Pólya's conjecture holds:
   $$k \leq (1 + \epsilon) \frac{|\Omega|\omega(n)}{(2\pi)^n} \lambda_k(\Omega)^{n/2}$$
   For convex domains, the threshold $\Lambda$ can be taken smaller. The paper notes that this reduces the verification of the conjecture for large eigenvalues to a computational problem for the finite set of eigenvalues below $\Lambda$.

3. New Classes of Domains Satisfying Pólya's Conjecture:
   The paper identifies new classes of domains in all dimensions $n \geq 2$ that satisfy the full Pólya conjecture, even if they possess irregular shapes or boundaries:

   • Strip-Tiling Domains: The authors show that strip-tiling domains satisfy an inequality strictly better than Pólya's conjecture (Theorem 1.10).
   • Domains with Removed Cubes: If a strip-tiling domain (or a product of a Pólya-satisfying domain and an interval) has a collection of disjoint cubes removed, the conjecture holds for the remaining domain provided the surface area of the "side face" of the original domain is sufficiently large relative to the total surface area of the removed cubes (Theorem 1.15).
   • Concrete Examples: This includes triangles in $\mathbb{R}^2$ (Corollary 1.12) and various domains formed by removing specific sequences of cubes from strip-tiling domains (Figure 2).

4. Improved Estimates for Triangles:
   For any triangle in $\mathbb{R}^2$, the paper establishes a sharper bound than the classical Pólya result:
   $$k \leq \frac{|\Omega|}{4\pi} \lambda_k(\Omega) - \frac{\max\{|ab|, |bc|, |ca|\}}{12\pi} \lambda_k(\Omega)^{1/2}$$

Significance and Claims
The paper claims that its primary significance lies in providing a quantitative and explicit approach to Pólya's conjecture. By establishing uniform remainder estimates with explicit constants, the authors reduce the problem of verifying the conjecture for large eigenvalues to a finite computational check.

The authors emphasize that their method yields a new proof of the Courant-type remainder estimate without using Neumann eigenvalues, which is a departure from classical techniques. Furthermore, the identification of strip-tiling domains and their perturbations (removing cubes) as classes satisfying Pólya's conjecture extends the known validity of the conjecture beyond tiling domains and simple shapes like balls and sectors. The paper modestly states that while it reduces the $\epsilon$-loss version to a computational problem, the full conjecture for general Lipschitz domains remains open, contingent on obtaining better lower bounds for the counting function of the complement domain $T \setminus \Omega$.