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Counting $2 \times 2$ integer matrices with a given determinant

The paper establishes an asymptotic formula with a quantitatively improved error term for the number of $2 \times 2$ integer matrices with entries in $[-N, N]$ and determinant $h$ , demonstrating square-root cancellation when $h \leq N$ and confirming the near-optimality of the error bound for large $h$ .

Original authors: Jonathan Chapman, Akshat Mudgal

Published 2026-05-19

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CC BY 4.0

Original authors: Jonathan Chapman, Akshat Mudgal

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 have a giant grid of integers, stretching from $-N$ to $N$ in every direction. Now, picture yourself trying to find specific 2x2 squares made of these numbers. But there's a catch: the "determinant" of your square (a specific calculation you do with the four numbers: $ad - bc$) must equal a specific target number, let's call it $h$ .

This paper is essentially a very precise counting game. The authors, Jonathan Chapman and Akshat Mudgal, are asking: How many of these special squares exist?

Here is the breakdown of their work using simple analogies:

1. The Main Goal: Counting the Squares

Think of the set of all possible 2x2 matrices as a massive, 4-dimensional warehouse filled with boxes. Each box contains four numbers. The authors want to count how many boxes in this warehouse have a "magic number" (the determinant) equal to $h$ .

They call this count $T(h, N)$ .

The "Main Term" (The Prediction): Before counting, mathematicians have a rough guess based on probability. It's like saying, "If you throw darts at a board, you expect to hit the bullseye about 16 times out of every 100 throws." The paper confirms that the number of squares is roughly proportional to the size of the warehouse ( $N^2$ ) and how many ways the target number $h$ can be broken down into factors.
The "Error Term" (The Surprise): The real world is messy. The actual count rarely matches the prediction perfectly. The difference between the prediction and the real count is called the "error term." The whole point of this paper is to figure out exactly how big this error can be.

2. The Big Improvement: Sharper Glasses

Previous researchers (like Afifurrahman and Ganguly–Guria) had looked at this problem with slightly blurry glasses. They could estimate the error, but their estimates were a bit loose, especially when the target number $h$ was large.

Chapman and Mudgal put on a new pair of glasses. They developed a method that is:

Simpler: They didn't need the most complex, heavy machinery (like deep spectral methods) that others used. They used "elementary" number theory, which is like solving a puzzle with basic logic rather than a supercomputer.
More Accurate: They proved that the error term is much smaller than previously thought. Specifically, when the target number $h$ is small (smaller than the size of the warehouse, $N$ ), the error is tiny—roughly the size of $N$ itself. This is a "square-root cancellation," which is a very desirable result in math, meaning the noise cancels itself out very efficiently.

3. The "Too Big" Problem: When the Target is Huge

The paper also looks at what happens when the target number $h$ is enormous (specifically, when $h$ is much larger than $N$ ).

The Analogy: Imagine trying to find a specific grain of sand on a beach. If the beach is small ( $N$ ) and you are looking for a grain that is supposed to be huge ( $h$ ), the rules change.
The Discovery: The authors found that when $h$ gets very large, the "error" doesn't stay small anymore. In fact, the error becomes as big as the target number $h$ itself.
Why it matters: This tells us that the standard "prediction formula" (the main term) stops working well when $h$ is too big. The "noise" overwhelms the signal. The paper proves that in this specific range, you simply cannot get a better estimate than what they found; the error is unavoidable and roughly the size of $h$ .

4. The "Zero" Case

The paper briefly mentions what happens if the target number is zero ( $h=0$ ). This is like looking for squares where the calculation results in zero. Because zero has special symmetries (many different combinations can result in zero), the math is actually easier here, and the count follows a slightly different, well-known pattern involving logarithms.

Summary of the "Takeaway"

What they did: They counted 2x2 integer matrices with a fixed determinant inside a bounded range.
How they did it: They used a clever, elementary method to break the problem down into counting points on lines and checking divisibility rules.
The Result: They gave a much tighter, more accurate formula for the count.
- If the target number is small, the count is very predictable, and the error is small.
- If the target number is huge, the error grows large, and the standard prediction formula hits a wall.

In short, they cleaned up the math on this specific counting problem, showing exactly how precise we can be and where the limits of that precision lie.

Technical Summary: Counting 2 × 2 Integer Matrices with a Given Determinant

Problem Statement
The paper addresses the problem of counting $2 \times 2$ integer matrices with a fixed determinant $h$ whose entries are bounded in absolute value by $N$ . Specifically, let $D_h = \{(a, b, c, d) \in \mathbb{Z}^4 : ad - bc = h\}$ . The authors define $T(h, N)$ as the cardinality of the intersection $D_h \cap [-N, N]^4$ . The goal is to establish an asymptotic formula for $T(h, N)$ as $N \to \infty$ , valid for $1 \le h \le 2N^2$ , and to quantify the error term $E_{\ell^\infty}(h, N)$ in the expansion:
$T(h, N) = \frac{16}{\zeta(2)}N^2 \left( \sum_{d|h} \frac{1}{d} \right) + E_{\ell^\infty}(h, N).$
This problem is situated within the broader context of counting integer solutions to quadratic equations and is related to the binary additive divisor problem and the distribution of points on modular hyperbolas.

Methodology
The authors employ a completely elementary approach, avoiding the deep spectral methods or estimates of Weil sums used in prior works. The methodology proceeds through the following steps:

Symmetry Reduction and Reformulation: The counting problem is reduced by exploiting symmetries of the matrix entries and excluding trivial cases (where entries are zero). The sum is reorganized based on the greatest common divisor of the coefficients in the linear Diophantine equation $ux - vy = h/d$.
Approximation via Uniformization: The authors introduce an auxiliary counting function $\tilde{r}_{u,v}(n)$ which counts solutions to a congruence condition $y \equiv -nv^{-1} \pmod u$ within a slightly relaxed interval. They demonstrate that the difference between the exact count $r_{u,v}(n)$ and the approximate count $\tilde{r}_{u,v}(n)$ is negligible for the purposes of the main theorem, provided $h$ is not excessively large relative to $N$ .
Arithmetic Progression Analysis: The problem is reduced to estimating the number of coprime pairs $(u, v)$ $(u, v)$ and counting integers in arithmetic progressions. The authors utilize standard lemmas from elementary number theory, including:
- The distribution of coprime residues in intervals (Lemma 2.4).
- Bounds on the number of solutions to congruences of the form $t^m \equiv z \pmod y$ (Lemma 2.2).
- Estimates for sums involving the greatest common divisor $(y, m)$ (Lemma 2.3).
Error Term Estimation: By carefully bounding the error introduced by the uniformization step and the main term summation, the authors derive the final asymptotic formula.

Key Results

Theorem 1.1 (Main Asymptotic Formula): For any $h, N \in \mathbb{N}$ with $1 \le h \le 2N^2$ , the authors prove:
$T(h, N) = \frac{16}{\zeta(2)}N^2 \left( \sum_{d|h} \frac{1}{d} \right) + O_\varepsilon(N^\varepsilon(N + h)).$
This result quantitatively improves upon previous bounds by Afifurrahman ( $O_\varepsilon(N^\varepsilon(h + N^{5/3}))$ ) and Ganguly–Guria ( $O_\varepsilon(h^\theta N^{3/2+\varepsilon})$ ). Crucially, in the range $h \le N$ , the error term satisfies $E_{\ell^\infty}(h, N) \ll_\varepsilon N^{1+\varepsilon}$ , which represents square-root cancellation and aligns with conjectured bounds for related divisor problems.
Theorem 1.2 (Lower Bound on Error): The authors establish that for $N^{1+\delta} \le h \le 2N^2$ (where $\delta \in (0, 1]$ ), the error term satisfies:
$E_{\ell^\infty}(h, N) \gg h.$
This demonstrates that when $h$ is large (specifically $h \ge N^{1+\delta}$ ), the error term is of the order of $h$ , which is comparable to the main term's magnitude in certain regimes. This implies that the conjectured bound $O_\varepsilon(N^{1+\varepsilon})$ cannot hold uniformly for all $h$ in the range $1 \le h \le 2N^2$ .

Significance and Claims
The paper claims to provide a "short and completely elementary" proof that significantly improves the known upper bounds for the error term in counting integer matrices with fixed determinant.

Improvement over Prior Work: The result supersedes recent work by Afifurrahman and Ganguly–Guria, offering a tighter error bound that confirms an analogue of the conjectured bounds for the binary additive divisor problem in the setting of $\ell^\infty$ -bounded matrices.
Optimality of Bounds: The lower bound in Theorem 1.2 suggests that the error term derived in Theorem 1.1 is essentially of the correct order when $h$ is large ( $h \gg N$ ). The authors note that for $h \gg N^2$ , the error term matches the order of the main term, indicating a shift in the nature of the asymptotic behavior, though they do not provide a new main term for this specific regime.
Independence: The authors acknowledge that Dhanda–Haynes–Prasala independently proved Theorem 1.1.

The paper does not propose new applications or future experimental directions but frames its contribution as a resolution to a specific quantitative problem in analytic number theory using elementary techniques, thereby clarifying the limits of current estimates for this class of problems.