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Local square mean in the hyperbolic circle problem

This paper improves the known pointwise error bound for the hyperbolic circle problem to $e^{(9/14+\epsilon)R}$ by establishing a stronger estimate for the local $L^2$ -norm of the error term, though this result remains weaker than the local average bound previously achieved by Petridis and Risager.

Original authors: András Biró

Published 2026-05-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: András Biró

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

The Big Picture: Counting Stars in a Curved Room

Imagine you are standing in a very strange, curved room (mathematicians call this the upper half-plane). The rules of geometry here are different from the flat floor you are used to. In this room, "straight lines" curve, and distances work differently.

Now, imagine there is a pattern of invisible tiles covering the entire floor, generated by a specific set of rules (a Fuchsian group). If you stand at a specific spot ( $z$ ) and look around, you can see copies of yourself scattered throughout the room. These are the "orbit" points.

The Problem:
You draw a giant circle around yourself with a radius $R$ . The Hyperbolic Circle Problem asks a simple question: How many of these "copies" of me are inside this circle?

Mathematicians have known for a long time that the answer is roughly proportional to the area of the circle. However, the count isn't exactly the area; there is a "wobble" or an error term.

The old, standard guess for how big this wobble is was roughly $R^{2/3}$ (about 0.66).
This means if the circle gets huge, the error grows, but we want to know exactly how fast.

The Breakthrough: Smoothing the Rough Edges

In this paper, the author, András Biró, doesn't try to fix the wobble for just one specific spot in the room. Instead, he asks: What happens if we look at the wobble for a whole neighborhood of spots at once and average them out?

Think of it like this:

The Old Way (Pointwise): You try to measure the height of a single, jagged mountain peak. It's very bumpy and hard to predict exactly.
The New Way (Local Average): You take a wide, soft blanket and lay it over a whole range of mountains. You measure the average height of the blanket. Because the high peaks and low valleys cancel each other out, the average is much smoother and easier to predict.

What Did They Actually Do?

The Setup: They took a specific, well-known pattern of tiles (related to the integers, called $PSL_2(\mathbb{Z})$ ).
The Smoothing: They used a mathematical "blender" (a technique involving smooth functions and spectral methods) to mix the counts from nearby points. This removes the extreme, jagged spikes in the error.
The Result: They calculated the "average squared error" (a way of measuring the typical size of the wobble) for this neighborhood.

The Magic Number:

The old limit was $R^{2/3}$ (0.66).
A previous team (Petridis and Risager) showed that if you average perfectly, you can get down to $R^{7/12}$ (about 0.58).
Biró's Result: He proved that for the "local square mean" (a specific type of average), the error is bounded by $R^{9/14}$ (approximately 0.64).

Why is this important?
$9/14$ (0.64) is smaller than $2/3$ (0.66). This means that when you look at the problem through this "average lens," the error is actually smaller than we thought it would be if we looked at a single point. It's a step forward, even though it doesn't quite reach the 0.58 limit of the perfect average.

The Secret Weapon: Quadratic Forms and "Pairs of Shoes"

To get this result, the author had to solve a very tricky counting problem involving quadratic forms.

Imagine you have pairs of shoes (two quadratic equations).

Each shoe has a "size" (discriminant).
The two shoes in a pair have a specific "relationship" (codiscriminant).
The author needed to count how many unique pairs of shoes exist that fit specific size and relationship rules.

He developed a new way to estimate the number of these pairs without needing an exact formula (which is too hard to find). He proved a "safety limit" (an upper bound) for how many of these pairs can exist. This safety limit was crucial for proving that the "wobble" in the circle problem stays within the new, tighter bounds.

The "Blender" Technique (The Proof Strategy)

The paper uses a clever trick called smoothing.

Instead of counting the points in a circle of radius $R$ directly, the author counts points in circles of radius $R, R-d, R-2d, \dots$ and mixes them together with specific weights (like a recipe).
This mixing cancels out the "noise" (the non-hyperbolic elements) and leaves only the "signal" (the hyperbolic elements).
By choosing the right size for the "mixing parameter" ( $d$ ), he balanced the error terms to get the best possible result ( $R^{9/14}$ ).

Summary

The Goal: Understand how many points fall into a hyperbolic circle.
The Challenge: The count is never perfect; there is an error.
The Innovation: Instead of looking at one point, the author looked at a small neighborhood and averaged the errors.
The Result: He proved that this average error is smaller than the old worst-case estimate ( $9/14$ is better than $2/3$ ).
The Method: He used a "blender" to smooth the data and a new counting method for pairs of mathematical "shoes" (quadratic forms) to keep the numbers in check.

In short, the paper shows that if you don't stare too hard at a single point, but instead look at the neighborhood as a whole, the chaotic wobble of the hyperbolic circle problem becomes much more predictable and smaller.

Technical Summary: Local Square Mean in the Hyperbolic Circle Problem

1. Problem Statement
The paper addresses the hyperbolic circle problem for a finite volume Fuchsian group $\Gamma \subseteq \text{PSL}_2(\mathbb{R})$ . Specifically, it focuses on estimating the number of elements in the $\Gamma$ -orbit of a point $z$ within a hyperbolic circle of radius $R$ centered at $w$ . For the specific case where $\Gamma = \text{PSL}_2(\mathbb{Z})$ and $z=w$ , the counting function is defined as:
$N(z, X) := |\{ \gamma \in \Gamma : 4u(\gamma z, z) + 2 \le X \}|$
where $u(z, w) = \frac{|z-w|^2}{4 \text{Im}z \text{Im}w}$ is related to the hyperbolic distance $\rho$ by $1+2u = \cosh \rho$ .

The classical pointwise bound for the error term $E(z, X) = N(z, X) - 3X$ is known to be $O_z(X^{2/3})$ , a result attributed to Selberg. While Petridis and Risager [P-R] improved the local average of this error term to $O(X^{7/12+\epsilon})$ by averaging over $z$ , the pointwise bound had not been improved for any group. This paper aims to improve the estimate for the local $L^2$ -norm (the square mean) of the error term.

2. Main Result
The primary theorem establishes an improved bound for the local $L^2$ -norm of the error term over a compact set $\Omega$ within the standard fundamental domain $F$ of $\text{PSL}_2(\mathbb{Z})$ .

Theorem 1.1: Let $\Gamma = \text{PSL}_2(\mathbb{Z})$ , let $F$ be the standard fundamental domain, and let $\Omega \subseteq F$ be a compact set. Then for any $\epsilon > 0$ :
$\left( \int_{\Omega} (N(z, X) - 3X)^2 d\mu_z \right)^{1/2} = O_{\Omega, \epsilon} \left( X^{9/14 + \epsilon} \right)$
The exponent $9/14 \approx 0.642$ is strictly better than the pointwise bound exponent $2/3 \approx 0.666$ , though it remains weaker than the $7/12 \approx 0.583$ exponent achieved by Petridis and Risager for the linear average.

3. Methodology
The proof employs a combination of spectral theory, arithmetic geometry, and analytic estimates.

Smoothing and Spectral Decomposition: The authors introduce a smoothed counting function $N_{d,J}(z, X)$ using a linear combination of $N(z, X-jd\tau)$ with binomial coefficients. This smoothing allows the application of spectral methods. The error term is decomposed into contributions from hyperbolic and non-hyperbolic elements of $\Gamma$ . The non-hyperbolic contribution is shown to be negligible ( $O(X^{1/2+\epsilon})$ ).
Reduction to Automorphic Inner Products: The core of the problem reduces to estimating the square integral of the hyperbolic contribution. Using the spectral expansion of automorphic functions, this integral is expressed in terms of inner products of sums of the form $M_{t,m}(z) = \sum_{\gamma \in \Gamma_t} m(u(z, \gamma z))$ , where $\Gamma_t$ is the set of elements with trace $t$ .
Connection to Quadratic Forms: A key technical innovation (Lemma 2.2) relates the inner product of these automorphic functions to arithmetic quantities. Specifically, the integral is expressed as a sum involving:
1. $E_{t_1, t_2}$ : A sum related to the logarithms of norms of generators of centralizers of pairs of hyperbolic elements with the same fixed points.
2. $h(d_1, d_2, t)$ : The number of $\text{SL}_2(\mathbb{Z})$ -equivalence classes of pairs of quadratic forms $(Q_1, Q_2)$ with discriminants $d_1, d_2$ and a specific "codiscriminant" $t$ .
Arithmetic Bounds: Since explicit formulas for $h(d_1, d_2, t)$ are only known under restrictive conditions, the authors prove a general upper bound for $h(d_1, d_2, t)$ (Lemma 3.1) involving the number of divisors and the square-free part of the parameters.
Analytic Estimates: The proof involves detailed analysis of special functions ( $I$ and $J$ functions) arising from the inner product formulas. These functions depend on parameters related to the traces and the smoothing scale. The authors utilize Taylor expansions, Faà di Bruno's formula for higher derivatives, and careful case analysis based on the relative sizes of the parameters to bound the sums.
Optimization: By balancing the error term from the spectral approximation ( $O(X/\sqrt{d})$ ) against the bound derived from the arithmetic sums ( $O(d^{5/2} X^{-1/2})$ ), the parameter $d$ is chosen as $X^{5/7}$ , yielding the final exponent $9/14$ .

4. Key Contributions

Improved $L^2$ Bound: The paper provides the first improvement over the $X^{2/3}$ pointwise bound for the hyperbolic circle problem in the context of the local $L^2$ -norm, achieving $X^{9/14+\epsilon}$ .
Arithmetic Tool: The derivation of a general upper bound for the class number of pairs of quadratic forms $h(d_1, d_2, t)$ (Lemma 3.1) without restrictive conditions on the parameters.
Spectral-Arithmetic Bridge: The explicit formula (Lemma 2.2) connecting the inner product of automorphic functions associated with trace sets to the arithmetic class numbers of quadratic form pairs.

5. Significance and Limitations
The significance of the result lies in demonstrating that averaging the error term locally in the $L^2$ -sense yields a better exponent than the worst-case pointwise bound, bridging the gap between the pointwise $X^{2/3}$ and the linear average $X^{7/12}$ .

The authors note several limitations and directions for future work:

Group Restriction: The current proof relies heavily on arithmetic tools specific to $\text{PSL}_2(\mathbb{Z})$ . Extending this result to arbitrary finite volume Fuchsian groups with an exponent better than $2/3$ remains an open problem, though the linear average result was previously extended by [B1].
Explicit Formulas: The authors suggest that the bound might be further improved if explicit formulas for $h(d_1, d_2, t)$ were available, rather than relying on the general upper bound derived in this paper.
Comparison: The result is explicitly stated to be weaker than the $X^{7/12}$ bound of Petridis and Risager for the linear average, but stronger than the pointwise bound.