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On the distance problem over finite p-adic rings

This paper resolves the distance problem over finite p-adic rings by establishing sharp results in odd dimensions, clarifying the conjecture in even dimensions with a new group-theoretic approach that proves the $4/3$ -parallel result in two dimensions, and providing a simpler, more flexible framework than previous methods to analyze geometric configurations in this setting.

Original authors: Thang Pham, Boqing Xue

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Thang Pham, Boqing Xue

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 throwing a massive party in a city built on a strange, layered grid. This isn't your normal city; it's a "p-adic ring" city, where every address is a number that can be peeled back into layers of smaller, nested neighborhoods. The distance between two guests isn't just a straight line; it's calculated by a special formula that wraps around, like a video game world where walking off the right edge makes you appear on the left.

The big question the authors, Thang Pham and Boqing Xue, are asking is: How many guests do you need at this party to guarantee that you can find every possible distance between them?

If you have a tiny crowd, you might only see a few distances. But if the crowd gets big enough, the "distance set" (the collection of all distances found between pairs of people) should eventually cover every possible distance the city allows. The paper is a detective story trying to figure out exactly how big that crowd needs to be.

The Odd-Dimension Mystery: A Perfect Fit

In cities with an odd number of dimensions (like 3D, 5D, etc.), the authors found a very precise answer. They proved that if your crowd density reaches a specific threshold—roughly $p^{-(n+1)/2}$ (where $p$ is a prime number defining the city's size)—you are guaranteed to see almost all distances.

Think of it like filling a bucket. If you pour in just a little water, the bottom isn't covered. But once you hit the exact line marked by their formula, the bucket is full. They didn't just guess this; they proved it mathematically. In fact, they showed that if you try to use fewer guests, you can build a "trick" party where the distances are missing, proving their number is the absolute minimum needed. It's a tight, perfect fit.

The Even-Dimension Puzzle: A Clever Conjecture

Things get trickier in even dimensions (like 2D or 4D). Here, the math gets slippery. The authors don't claim to have solved the whole puzzle yet. Instead, they propose a conjecture (a strong, well-supported guess) that the crowd needs to be slightly smaller than in the odd case—specifically around $p^{-n/2}$ .

To back up this guess, they built specific examples of "trick parties" in even dimensions where the distances fail to cover everything if the crowd is too small. These examples act like proof-of-concept models, showing that their guess makes sense, even if they haven't written the final, unbreakable proof for every single case yet.

The Two-Dimensional Breakthrough: Cracking the Code

The most exciting part of the story happens in two dimensions (a flat plane). For a long time, mathematicians wondered if a specific "magic number" for crowd size existed here. In the simpler world of finite fields (a flat, single-layer version of the city), the answer was known to be $4/3$ . But in this complex, multi-layered p-adic city, no one knew for sure.

The authors answered a question posed by mathematician Alex Iosevich: Does this $4/3$ rule still hold in the complex city?

They said yes, but specifically for the two-dimensional case. By using a clever mix of group theory (studying how shapes rotate and move) and new mathematical tools called "restriction estimates" (which act like filters to sort out the noise), they proved that if your crowd density is at least $p^{-2/3}$ in two dimensions, you will definitely find all the distances. They didn't just guess; they built a rigorous argument that holds up under scrutiny. However, this specific proof applies to 2D, not all even dimensions.

The "Transfer Principle": Borrowing Answers from Simpler Worlds

One of the paper's coolest tricks is a "transfer principle." Imagine you have a hard problem in the complex, multi-layered city, but you know the answer in a simple, single-layer city. The authors developed a method to "lift" the answer from the simple city to the complex one.

They showed that if you can count the number of "isosceles triangles" (triangles with two equal sides) in a simple finite field, you can use that information to figure out distance problems in the complex p-adic rings. This allowed them to take existing results from other mathematicians and apply them to these new, more complex settings, proving results for 2D and 4D cases without having to start from scratch.

What They Ruled Out

The paper is very clear about what doesn't work.

They explicitly show that you cannot simply use the old, simpler methods from finite fields to solve these complex ring problems. The old methods fail to find the "uniform density" needed because they get confused by the layers of the p-adic rings.
Crucially, they did not rule out the idea that the density threshold could be independent of the prime number $p$ . On the contrary, a major novelty of their work is that their results provide a uniform density that is independent of the number of layers ( $r$ ) in the ring. Their methods successfully find a threshold that works consistently regardless of how deep the "city" goes, which was a significant hurdle for previous approaches.

The Verdict

The authors are proven correct about the odd dimensions and the specific 2D case ( $4/3$ rule). They are highly confident in their conjecture for even dimensions, supported by strong examples, but they acknowledge it is still a conjecture until fully proven for all cases.

They also introduced a new, simpler way to do the math compared to previous researchers (like Ben Lichtin). While Lichtin's method was like climbing a mountain with a heavy backpack (using complex second-order calculations), the authors' method is like taking a direct path (using first-order calculations and polynomial congruences). It's faster, more flexible, and gets the job done with less fuss.

In short, they mapped out the exact crowd size needed to see every distance in odd dimensions, cracked the code for the 2D case, and provided a strong roadmap for the rest of the even dimensions, all while showing us a simpler way to walk the mathematical path.

Technical Summary: On the distance problem over finite p-adic rings

Problem Statement
This paper investigates the Erdős–Falconer distance problem within the setting of finite $p$ -adic rings, denoted as $(\mathbb{Z}/p^r\mathbb{Z})^n$ . The central question (Question 1.1) seeks the smallest density threshold $\delta \in (0, 1)$ , independent of the exponent $r$ , such that for any subset $E \subset (\mathbb{Z}/p^r\mathbb{Z})^n$ with density $\delta_E \ge \delta$ , the distance set $\Delta_{n,r}(E) = \{ \|x-y\| : x, y \in E \}$ covers a significant portion of the ring (specifically, $|\Delta_{n,r}(E)| \gg p^r$ ). The distance function is defined by the standard quadratic form $\|x-y\| = \sum (x_i - y_i)^2 \pmod{p^r}$ , though the authors also consider generalized polynomial distance functions.

The problem extends previous work over finite fields ( $r=1$ ), where the threshold is known to be sharp at $p^{-(n+1)/2}$ for odd dimensions and conjectured to be $p^{-n/2}$ for even dimensions. In the $p$ -adic setting ( $r > 1$ ), prior results by Lichtin (2019, 2023) established thresholds but relied on complex $p$ -adic analysis and Hessian matrix computations. The paper aims to provide a simpler, more flexible approach and to clarify the conjecture in even dimensions.

Methodology
The authors employ a combination of Fourier analysis over finite rings, polynomial congruences, and group-theoretic arguments. Key methodological components include:

Fourier Analysis and Exponential Sums: The core of the distance problem analysis relies on estimating the number of pairs $(x, y)$ such that $F(x-y) \equiv j \pmod{p^r}$ . This is achieved by decomposing the counting function into a main term and an error term using the Fourier transform. The error term is controlled by bounding exponential sums associated with the level sets of the polynomial $F$ .
Hensel's Lemma and Jacobian Matrices: Unlike Lichtin's approach, which utilized second-order Hessian matrices to detect cancellations in exponential sums, this paper relies primarily on first-order Jacobian matrices and Hensel's Lemma. This simplification allows for a more direct lifting of results from the residue field $\mathbb{F}_p$ to the ring $\mathbb{Z}/p^r\mathbb{Z}$ .
Extension Estimates: For the two-dimensional case, the authors develop new restriction-type estimates associated with circles and orbits under the action of the special orthogonal group $SO_2(\mathbb{Z}/p^r\mathbb{Z})$ . These estimates are derived using group-theoretic arguments to reduce the problem to extension questions for specific varieties.
Transfer Principle: A significant methodological contribution is the formulation of an $F_p$ -to- $\mathbb{Z}/p^r\mathbb{Z}$ transfer principle. This principle separates combinatorial inputs (isosceles energy estimates over finite fields) from higher-conductor analytic analysis, allowing known finite field results to be lifted to arbitrary $p$ -adic rings.

Key Contributions and Results

Generalized Distance Sets (Odd Dimensions):
Theorem 1.4 establishes a general condition for polynomials $F(x)$ under which the distance set is large. If the density $\delta_{E_1, E_2}$ exceeds $C p^{-(n+1)/2}$ , the distance set contains all units. The authors prove that this threshold is essentially sharp in odd dimensions (Example 8.1). This result generalizes and simplifies previous work by Lichtin.
Even Dimensions and the 4/3-Parallel Result:
In two dimensions, the paper addresses a question raised by Alex Iosevich regarding the existence of a $4/3$ -parallel result (analogous to the finite field case). Theorem 1.8 proves that if $\delta_E \gg p^{-2/3}$ , then $|\Delta_{2,r}(E)| \gg p^r$ . This is achieved by developing new extension theorems for circles and orbits in the $p$ -adic setting (Theorems 4.1–4.3).
Furthermore, Theorem 1.9 and Corollary 1.10 provide a large family of sets for which the distance conjecture holds in even dimensions, specifically those where the density of fibers over the projection to $\mathbb{F}_p$ is controlled.
Transfer Principle and Higher Dimensions:
The paper introduces a transfer principle (Theorem 6.1) that lifts isosceles energy estimates from finite fields to $p$ -adic rings. Applying this to existing results in dimensions 2 and 4, the authors derive Theorems 1.11 and 1.12, providing density thresholds for these dimensions that are independent of $r$ .
Geometric and Graph Configurations:
The authors extend their distance results to the existence of various geometric configurations, including rectangles (Theorem 1.13), 4-cycles (Theorem 1.14), $k$ -chains (Theorem 1.15), and pinned trees (Theorem 1.16). These results improve upon Lichtin's density thresholds for $k$ -chains, reducing the required density from $p^{-(n+1)/3}$ to $p^{-(n+1)/2}$ .

Significance and Claims
The paper claims that its primary novelty lies in the simplicity and flexibility of the approach compared to Lichtin's method. By avoiding second-order Hessian computations and relying on first-order Jacobian analysis and polynomial congruences, the authors provide a more accessible framework for studying distance problems in $p$ -adic rings.

The authors explicitly state that their results are "essentially sharp in odd dimensions" and that they "clarify the conjecture in even dimensions" by providing supporting examples and proving the $4/3$ -parallel result in two dimensions. The work also answers a specific question posed by Alex Iosevich regarding the existence of the $4/3$ result in the $p$ -adic setting. The paper positions itself as an improvement and extension of recent work by Ben Lichtin, offering a unified approach that handles generalized polynomials and various geometric configurations with uniform density thresholds independent of the ring exponent $r$ .