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Lattice point counting in Cygan--Korányi balls on Heisenberg groups

This paper improves the known upper bound for the error term in the lattice point counting problem for Cygan–Korányi balls on the Heisenberg group $\mathbb{H}^q$ (for $q \ge 4$ ) to $O(t^{2q-1+241/753})$ using Landau's formula and van der Corput's derivative tests, marking the first progress toward Gath's conjecture of an optimal $O(t^{2q-1})$ bound.

Original authors: Sheng-Chen Mao, Sibei Yang

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sheng-Chen Mao, Sibei Yang

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 a cosmic baker trying to count how many chocolate chips fit inside a giant, invisible, squishy cookie. In the flat, boring world of a kitchen counter (what mathematicians call "Euclidean space"), this is the famous "Gauss circle problem." You draw a circle, count the chips, and subtract the expected amount. The difference is your "error." For a long time, mathematicians have been arguing about exactly how big that error can get.

Now, imagine taking that cookie and twisting it into a weird, non-flat shape where the rules of distance change depending on how you move. This is the Heisenberg group, a mathematical universe that behaves like a twisted, non-commutative dance floor. In this world, the "cookie" isn't a circle; it's a Cygan–Kor´anyi ball. It looks like a sphere, but if you stretch it or squeeze it, it changes shape in a very specific, tricky way.

The big question the paper tackles is: How many "chips" (lattice points) fit inside this twisted ball as it grows huge?

The Great Twisty-Ball Mystery

Back in 2015, a team of mathematicians (Garg, Nevo, and Taylor) took a first bite at this problem. They figured out how to count the chips for small and medium-sized balls, but for the really big ones, their estimate was a bit loose. They said the error would grow at a certain speed, but it felt like they were guessing the speed of a car by looking at its shadow.

Then, in 2022, a mathematician named Gath stepped in. He sharpened the tools and found a better estimate. He also made a bold guess (a conjecture): he believed the error grows at a very specific, "optimal" speed, which he called $2q - 1$ (where $q$ is a number describing the dimension of the space). Think of this as Gath betting that the error grows exactly as fast as a specific type of rocket, no faster, no slower.

The New Recipe: A Slice of Math

The authors of this paper, Mao and Yang, decided to test Gath's rocket bet. They didn't just look at the whole ball; they used a clever trick called slicing. Imagine slicing the twisted ball into thin, flat layers. Each layer looks a bit like a standard Euclidean circle, but with a twist.

They used a powerful mathematical tool called Landau's formula (which is like a recipe for counting chips in standard circles) to translate the problem from the weird twisted world back to the familiar flat world. This turned their problem into a massive sum of numbers that wiggle up and down (oscillatory sums).

Here is where it gets tricky. To count these wiggles, they had to use a technique called the Derivative Test. Imagine you are trying to predict the path of a bouncy ball. If you look at how fast it's moving (the first derivative), you get a rough idea. If you look at how fast it's accelerating (the second derivative), you get better. The authors had to look at the 5th and 6th derivatives—which is like analyzing the ball's "jerk," "snap," "crackle," and "pop" all at once to predict its path.

The Discovery: A Step Forward, Not a Finish Line

The authors ran their calculations and found something exciting, but not a total victory.

For the bigger dimensions ( $q \ge 4$ ): They proved that the error grows at a speed of $t^{2q - 1 + \frac{994}{753}}$ .
- Wait, what? That looks messy! The "optimal" speed Gath guessed was just $t^{2q-1}$ . The authors found a speed that is slightly slower than the worst-case scenario but slightly faster than Gath's perfect guess.
- The fraction $\frac{994}{753}$ (which is about 1.32) is the extra "wobble" they couldn't quite eliminate. It's like they managed to catch the rocket, but it's still wobbling a bit in the wind. They didn't prove Gath's exact number, but they got much closer than anyone else has before.
For the specific case of $q = 3$ : They found the error grows at $t^{\frac{16}{3}} \log t$ .
- This matches what Gath had already found, but their method was simpler and cleaner. They didn't improve the number here, but they showed their new "slicing" recipe works just as well.

What They Didn't Do (The "No" List)

It is important to know what this paper didn't do.

They did not prove Gath's conjecture is 100% true. They didn't reach the "optimal" speed of $2q-1$ exactly.
They did not solve the problem for the smallest dimension ( $q=2$ ). Their method actually got stuck there because the "wobble" in their math was too big to improve on previous results.
They did not simulate this on a computer. This is a pure math proof, built on logic and formulas, not a video game simulation.

The Verdict

The paper is a significant step forward. It's like the first team to climb a mountain and plant a flag at a high camp, proving the summit is reachable, even if they haven't stood on the very top yet. They showed that Gath's guess is likely in the right ballpark, but there is still a tiny bit of "noise" (the $\frac{994}{753}$ part) that needs to be smoothed out.

They used a mix of old-school number theory (Landau's formula) and modern harmonic analysis (the 6th Derivative Test) to show that the twisted ball problem is just as hard and fascinating as the original flat circle problem. The journey to the perfect answer continues, but thanks to Mao and Yang, we now have a better map for the next leg of the climb.

Technical Summary: Lattice Point Counting in Cygan–Kor´anyi Balls on Heisenberg Groups

Problem Statement
The paper addresses the lattice point counting problem within gauge balls on the Heisenberg group $H_q$ (where $q \ge 3$ is an integer). Specifically, it investigates the error term $E_q(t)$ associated with counting integer points in the Cygan–Kor´anyi ball $B_t = \{(v, w) \in H_q : (|v|^4 + w^2)^{1/4} \le t\}$ . The error term is defined as:
$E_q(t) = \#(\mathbb{Z}^{2q+1} \cap B_t) - \text{vol}(B_1)t^{2q+2}.$
This problem is presented as a non-commutative analogue of the Euclidean multidimensional sphere problem and is closely related to the classical Gauss circle problem. The central objective is to determine the optimal order of magnitude for $|E_q(t)|$ .

Context and Prior Work
Previous work by Garg, Nevo, and Taylor [8] established initial bounds, showing $|E_q(t)| \lesssim t^{2q}$ for $q \ge 3$ . In 2022, Gath [10] improved this upper bound to $t^{2q - 2/3}$ and derived a lower bound of order $t^{2q-1}(\log t)^{1/4}$ . Based on these findings, Gath conjectured that the optimal order for the error term is $2q - 1$ .

Methodology
The authors employ a method distinct from previous approaches (which utilized Poisson summation, spectral estimates, or hyperplane slicing with Bessel functions). Their strategy involves:

Reduction to Euclidean Error Terms: By utilizing a specific slicing technique, the problem is reduced to estimating a sum involving the lattice point error term $P_{2q}(r)$ for standard Euclidean balls in $\mathbb{R}^{2q}$ .
Landau's Formula: The authors apply Landau's formula (1924) for the error term of Euclidean lattice point counting. This transforms the problem into analyzing a main term involving the sawtooth function $\psi$ and a secondary term involving exponential sums.
Transformation to $\Psi$ -sums: The error term is expressed in terms of $\Psi$ -sums (specifically involving the function $\psi_k$ ), which are 1-periodic, uniformly bounded, and of uniformly bounded variation. These sums are structurally similar to those appearing in the Gauss circle problem.
Ste˘ckin's Inequality: The $\Psi$ -sums are converted into exponential sums using Ste˘ckin's inequality.
Van der Corput Derivative Tests: To bound the resulting exponential sums, the authors apply the 5-th and 6-th Derivative Tests of van der Corput. A key technical difficulty noted is that the phase function contains an additional parameter $k$ , and the summation range is broader than in the standard Gauss circle problem. This causes the 5-th derivative of the phase to vanish in certain intervals. To overcome this, the authors utilize a pair of derivative tests (orders 5 and 6) such that the non-vanishing intervals of one cover the zeros of the other.

Key Results
The paper establishes the following upper bounds for the error term $E_q(t)$ for $t \ge 10$ :

For $q \ge 4$ :
$|E_q(t)| \lesssim t^{2q - 1 + \frac{994}{753}}$
The exponent $\frac{994}{753} \approx 1.32005$ is derived from the 6-th Derivative Test. The authors note that this exponent is inferior to the potential $4\theta^* + \epsilon$ (where $\theta^* \approx 0.314483$ is the best known exponent for the Gauss circle problem), suggesting that further improvements might be possible by adapting methods from Li and Yang [23].
For $q = 3$ :
$|E_3(t)| \lesssim t^{16/3} \log t$
This bound is slightly coarser than the $q \ge 4$ case due to a loss of precision in the outer summation of Landau's formula. However, it recovers Gath's previous result up to a logarithmic factor.

Significance and Claims
The authors claim that this work provides the first progress towards Gath's conjecture (that the optimal order is $2q - 1$ ) for $q \ge 4$ . By achieving an exponent of $2q - 1 + \epsilon'$ (where $\epsilon' = 994/753 - 1 \approx 0.32$ ), the paper moves the upper bound closer to the conjectured optimal order of $2q - 1$ .

The paper emphasizes the intrinsic connection between the Cygan–Kor´anyi ball problem and the Gauss circle problem. The appearance of the specific exponent $\frac{497}{1506}$ (derived from the 6-th derivative test) in the intermediate estimates serves as evidence that the Cygan–Kor´anyi problem shares the same level of difficulty as the Gauss circle problem. The authors explicitly state that if the bound for the associated $\Psi$ -sum could be improved to match the conjectured optimal order of the Gauss circle problem ( $x^{1/4+\epsilon}$ ), their method would immediately verify Gath's conjecture for all $q \ge 4$ .