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Counting $2\times 2$ matrices with fixed determinant and bounded coefficients

This paper provides a new proof with an improved error term for the asymptotic count of $2\times 2$ matrices with fixed determinant $n$ and coefficients bounded by $X$ , while also demonstrating that the required bound $X \ge n^{1/2+\delta}$ is sharp by showing a different asymptotic behavior when $X=n^{1/2}$ .

Original authors: Kavita Dhanda, Alan Haynes, Silmi Prasala

Published 2026-04-01

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CC BY 4.0

Original authors: Kavita Dhanda, Alan Haynes, Silmi Prasala

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 master architect trying to build a specific type of 2x2 grid (a matrix) using only whole-number bricks. You have two strict rules for your construction:

The "Magic Number" Rule: When you multiply and subtract the bricks in a specific way (the determinant), the result must be a specific number, let's call it $n$ .
The "Size Limit" Rule: None of your bricks can be too big. They must all fit inside a box of size $X$ .

The big question mathematicians have been asking is: How many different ways can you build this grid if you make the box ( $X$ ) huge?

This paper by Kavita Dhanda, Alan Haynes, and Silmi Prasala is like a new, sharper blueprint for answering that question. Here is the breakdown in everyday terms:

1. The Old Map vs. The New GPS

For a long time, mathematicians knew the general shape of the answer. If you make the box ( $X$ ) infinitely large, the number of possible grids grows roughly like the area of the box ( $X^2$ ).

However, the "error" in the old maps was fuzzy. It was like saying, "There are about a million ways to build this, give or take a few thousand."

The Previous Best (Afifurrahman): In 2024, a mathematician named Afifurrahman drew a better map. He gave a formula that worked well, but the "give or take" part (the error term) was still a bit loose, especially when the target number $n$ was huge compared to the box size $X$ .
The New Map (This Paper): The authors in this paper have built a GPS. They didn't just prove the same result; they tightened the error term significantly. Their formula is much more precise. It tells you exactly how many grids exist, even when the box size $X$ is surprisingly small (as long as it's slightly bigger than the square root of $n$ ).

The Analogy: Imagine counting how many ways you can arrange a deck of cards to get a specific hand.

The old method said: "There are roughly $100$ ways, maybe plus or minus $20$."
This new method says: "There are exactly $100$ ways, plus or minus $2$."
Even better, this new method works even if you are only allowed to use a very small subset of the deck.

2. The "Square Root" Threshold

One of the most exciting discoveries in the paper is finding the limit of possibility.

The authors proved that their new, super-precise formula works perfectly down to a box size of roughly $\sqrt{n}$ (the square root of $n$ ).

Think of it like this: If your target number $n$ is $10,000$, the formula works great as long as your box size $X$ is bigger than $100$.
The Breakthrough: They proved that if you try to shrink the box any smaller than that (below $\sqrt{n}$ ), the formula breaks. The math changes completely.

3. The "Trap" at the Edge

To prove that the formula must break below the square root, the authors looked at a special case: when $n$ is a prime number (like 7, 13, 101) or the square of a prime (like 49, 169).

They set the box size $X$ to be exactly $\sqrt{n}$ .

The Result: When they counted the grids in this tiny box, the number they got was different from what the main formula predicted.
The Metaphor: Imagine you have a recipe for baking cookies that works perfectly for large batches. The authors showed that if you try to bake just a tiny batch (the "prime" case), the oven behaves differently, and you get a different number of cookies than the big-batch recipe predicts. This proves that the "big batch" recipe has a hard limit; it can't be used for the tiniest boxes.

Why Does This Matter?

You might ask, "Who cares about counting 2x2 grids?"

Cryptography: These grids are related to how we encrypt data. Understanding the "density" of these grids helps us understand the security of certain codes.
Number Theory: It helps mathematicians understand how numbers are distributed in space.
Precision: In science and engineering, knowing the exact error margin is crucial. If you are building a bridge or a satellite, you don't want a "rough estimate"; you want the "GPS precision" these authors provided.

Summary

This paper is a victory for precision.

It refines an existing formula to be much more accurate, especially when the target number is large.
It identifies the exact breaking point (the square root of $n$ ) where the rules of the game change.
It uses clever logic (like the "Pigeonhole Principle"—if you have more pigeons than holes, at least one hole must have two pigeons) to prove that you can't go any smaller than that breaking point without changing the math entirely.

In short: They found a better way to count, and they proved exactly where that counting method stops working.

1. Problem Statement

The paper addresses the problem of estimating the number of $2 \times 2$ integer matrices $A$ with a fixed non-zero determinant $n \in \mathbb{N}$ and coefficients bounded in absolute value by $X$ . Specifically, the authors study the counting function:
$S_{2, \|\cdot\|_\infty}(n, X) = \# \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{Z}^{2 \times 2} : \det(A) = n, \max(|a|, |b|, |c|, |d|) \le X \right\}$
where $\|\cdot\|_\infty$ denotes the sup-norm.

While asymptotic formulas for this problem exist for the Euclidean norm (based on rotational invariance) and for the specific case of $n=0$ , the behavior for the sup-norm with fixed $n \neq 0$ was less understood until recent work by M. Afifurrahman (2024). The authors aim to refine these estimates, particularly regarding the error term and the range of validity for $X$ relative to $n$ .

2. Methodology

The authors employ a combination of analytic number theory techniques and combinatorial arguments:

Decomposition of the Counting Function:
The authors decompose the set of matrices based on the relationship between the entries. They define sets $S_{leq}$ (where $|r| \le |q|$ ) and $S_{eq}$ (where $|r| = |q|$ ) to relate the total count $S(n, X)$ to sums over divisors.
$S(n, X) = 2S_{leq}(n, X) - S_{eq}(n, X)$
This reduces the problem to counting solutions to the linear Diophantine equation $ar - bq = n$ under specific bounds.
Divisor Summation and Modular Arithmetic:
The proof of Theorem 1 involves summing over divisors $d$ of $n$ . By setting $q = d q_d$ and $n = d n_d$ , the equation is transformed. The authors utilize the condition $(a, q) | n$ to restrict the summation over $a$ .
They analyze the number of solutions to $a r' - b q' = n'$ modulo $q'$ , utilizing properties of the Euler totient function $\phi$ and the Möbius function.
Error Term Analysis:
A significant portion of the proof involves bounding the "error" introduced when relaxing strict inequality constraints in the modular arithmetic steps. They use standard analytic number theory bounds involving $\sigma_0(n)$ (the number of divisors) and $\sigma_{-1}(n)$ (the sum of reciprocal divisors).
Pigeonhole Principle for Lower Bounds:
For Theorem 2, the authors use the Dirichlet Pigeonhole Principle. They construct a set of pairs $(a, q)$ within a specific range and map them to residues modulo $n$ . By showing the map is surjective and analyzing the collisions (where $ar - bq = \pm n$ ), they derive the asymptotic behavior for small $X$ .

3. Key Contributions and Results

Theorem 1: Improved Asymptotic Formula

The authors provide a new, shorter proof for the asymptotic formula of $S_{2, \|\cdot\|_\infty}(n, X)$ with a significantly improved error term compared to Afifurrahman's previous work.

Result:
For any $\epsilon > 0$ , $n \in \mathbb{N}$ , and $X \ge 1$ :
$S_{2, \|\cdot\|_\infty}(n, X) = \frac{96 \sigma_{-1}(n)}{\pi^2} X^2 + O_\epsilon \left( \sigma_0(n) X \log X + n X^\epsilon \right)$

Significance of the Error Term:

Uniformity: The error term is uniform in both $n$ and $X$ .
Range of Validity: The formula yields significant estimates even when $X$ is as small as $n^{1/2 + \delta}$ .
Improvement: When $X \gg n$ , the error term offers a "significant power savings" compared to Afifurrahman's $O(X^{5/3})$ and the subsequent improvement by Ganguly and Guria ( $O(n^\theta X^{3/2+\epsilon})$ ).

Theorem 2: Threshold Behavior and Counter-Example

The authors demonstrate that the exponent $1/2 + \delta$ in Theorem 1 is sharp. They prove that when $X$ drops to exactly $n^{1/2}$ (specifically for $n=p$ or $n=p^2$ where $p$ is prime), the main term changes, indicating that the formula in Theorem 1 fails to provide a correct asymptotic estimate in this regime.

Result:
For a prime $p$ :

$S_{2, \|\cdot\|_\infty}(p, p^{1/2}) = 4 \left( \frac{12}{\pi^2} - 1 \right) p + O(p^{1/2} \log p)$
$S_{2, \|\cdot\|_\infty}(p^2, p) = 4 \left( \frac{12}{\pi^2} - 1 \right) p^2 + O(p \log p)$

Key Observation:
The constant in the main term here is $4(\frac{12}{\pi^2} - 1)$ , which is distinct from the constant $\frac{96}{\pi^2}$ in Theorem 1. This proves that the asymptotic behavior undergoes a phase transition at $X \approx n^{1/2}$ .

4. Significance and Impact

Refinement of Existing Theory: The paper resolves open questions regarding the error terms in counting matrices with bounded coefficients, providing a tighter bound that is uniform in the determinant $n$ .
Sharpness of Exponents: By establishing Theorem 2, the authors prove that the threshold $X \sim n^{1/2}$ is a fundamental barrier. Below this threshold, the distribution of matrices is governed by different arithmetic properties (related to the "minimal denominator problem" in $p$ -adic numbers), rendering the standard main term invalid.
Methodological Efficiency: The proof of Theorem 1 is described as "short" and self-contained, offering a more accessible derivation than previous complex ergodic or spectral approaches.
Future Directions: The discrepancy in constants for $X = n^{1/2}$ suggests a deeper connection to $p$ -adic minimal denominator problems, opening a new avenue for research into higher powers of primes and more general arithmetic settings.

In summary, this paper establishes the most precise asymptotic formula currently known for $2 \times 2$ matrices with fixed determinant and bounded sup-norm coefficients, while rigorously defining the limits of this formula's applicability.