Counting solutions to the quadratic determinant equation

Technical Summary: Counting Solutions to the Quadratic Determinant Equation

Problem Statement
This paper addresses the problem of counting integer solutions to the quadratic determinant equation $x_1x_2 - x_3x_4 = h$ within a sharp box $[-N, N]^4$. Specifically, the authors analyze the quantity:
$$T(h, N) = |\{x \in \mathbb{Z}^4 : x_1x_2 - x_3x_4 = h, \ x_i \in [-N, N]\}|$$
where $h, N \in \mathbb{N}$ and $1 \leqslant h \leqslant N^2$. The study focuses on deriving an asymptotic formula for $T(h, N)$ with strong error terms, particularly in the regime where $h$ is close to $N^2$ (specifically $h = N^2 + O(N)$).

The authors contrast this "sharp cut-off" problem with two related, well-studied settings:

1. Smooth weights: Where the region is weighted by a smooth function, allowing for the application of the circle method (e.g., Heath-Brown's results).
2. Euclidean norm constraints: Where $x_1^2 + \dots + x_4^2 \leqslant N^2$, a setting where spectral theory of automorphic forms yields specific error bounds (e.g., Selberg's results).

The challenge lies in the fact that the sharp cut-off and the arbitrary nature of $h$ (relative to $N$) typically prevent the circle method from yielding asymptotic formulas with uniform error terms.

Methodology
The authors employ a hybrid approach combining combinatorial arguments in physical space with analytic number theory, specifically avoiding a direct reliance on the circle method's heuristic predictions for the sharp case.

1. Decomposition into Additive and Difference Types:
   The problem is reduced by analyzing solutions where variables are non-zero. The equation $x_1x_2 - x_3x_4 = h$ is decomposed based on the signs of the variables, leading to two primary sub-problems:

   • Additive-type: Solutions to $ax + by = h$ with $a,b,x,y \in [N]$.
   • Difference-type: Solutions to $ax - by = h$ with $a,b,x,y \in [N]$.

2. Discrete-to-Continuous Approximation:
   For the additive case, the authors fix the greatest common divisor $d = \gcd(x, y)$ and reduce the equation to $au + bv = h/d$ with $\gcd(u, v) = 1$. They express the number of solutions as a sum over $u, v$ of the length of an arithmetic progression.

   • The main term is derived by approximating the discrete sum with a continuous integral over the region defined by the constraints.
   • The coprimality condition is removed via Möbius inversion, introducing an auxiliary sum over $k$.
   • The resulting integrals are identified as the real density $\sigma_\infty(h/N^2)$, while the arithmetic sums correspond to the singular series $S_h$.

3. Error Term Analysis:
   The error terms arise from:

   • The discrepancy between the discrete count and the continuous integral (discretization error).
   • The sawtooth function $\psi(x) = x - \lfloor x \rfloor - 1/2$, which appears when counting lattice points in intervals.
   • The authors bound the sawtooth contributions using truncated Fourier expansions. This reduces the problem to estimating incomplete Kloosterman sums.
   • General Case ($h$ arbitrary): Using standard bounds for incomplete Kloosterman sums (via Weil bounds and Hooley's lemmas), the error term is shown to be $O_\varepsilon(N^{3/2+\varepsilon})$.

4. Special Case ($h \approx N^2$):
   When $h = N^2 + \Delta$ with $\Delta = O(N)$, the geometry of the solution space changes. The authors exploit an additional symmetry available in this specific regime.

   • They show that the intervals of summation for the variables become "complete" modulo $u$ up to a small error.
   • This allows the replacement of incomplete Kloosterman sums with Ramanujan sums.
   • By utilizing properties of Ramanujan sums and the triangle inequality, they bypass the need for the stronger Kloosterman sum bounds, achieving a significantly sharper error term of $O_\varepsilon(N^{1+\varepsilon})$.

Key Contributions and Results

• Theorem 1.1 (General Asymptotic): For any $h, N \in \mathbb{N}$, the authors prove:
  $$T(h, N) = \sigma_\infty(h/N^2) S_h N^2 + O_\varepsilon(N^{3/2+\varepsilon})$$
  where $\sigma_\infty$ is the real density and $S_h$ is the singular series. This confirms the circle method heuristic quantitatively despite the sharp cut-offs.

• Theorem 1.3 (Square-Root Cancellation): In the specific case where $h = N^2 + O(N)$, the error term is improved to:
  $$T(h, N) = \left(\frac{8}{\zeta(2)} - 4\right) \left(\sum_{d|h} \frac{1}{d}\right) N^2 + O_\varepsilon(N^{1+\varepsilon} + N^\varepsilon |\Delta|)$$
  This result exhibits square-root cancellation ($O(N^{1+\varepsilon})$), which is a significant improvement over the $O(N^{3/2+\varepsilon})$ bound.

• Explicit Formulas: The paper provides explicit calculations for the singular series $S_h = \zeta(2)^{-1} \sum_{d|h} d^{-1}$ and the real density $\sigma_\infty(\lambda)$ in terms of the dilogarithm function $\text{Li}_2$ and logarithmic terms, distinguishing cases for $\lambda \in (0, 1)$, $\lambda = 1$, and $\lambda \in (1, 2)$.

• Corollary 1.4: The authors confirm a speculation by Dhanda–Haynes–Prasala regarding the error term size when $h$ is a power of a prime (or generally $h = N^2 + O(N)$), showing the error is $O_\varepsilon(N^{1+\varepsilon})$.

Significance and Claims
The paper claims to resolve a speculation regarding the error term size for the quadratic determinant equation in the "critical" range where $h \approx N^2$.

• Comparison to Smooth Weights: While smooth-weight analogues (Heath-Brown) typically yield $O(N^{3/2+\varepsilon})$ errors, this work achieves the same strength for sharp cut-offs in the general case and improves to $O(N^{1+\varepsilon})$ in the symmetric case.
• Comparison to Spectral Theory: The authors note that for the $\ell^2$-norm constrained problem (Selberg's setting), the best known error bounds are of order $N^{4/3}$ or larger. In contrast, their result for the sharp box with $h \approx N^2$ achieves $O(N^{1+\varepsilon})$, which is significantly smaller than the main term size ($N^2$) and represents a stronger cancellation than currently known for the $\ell^2$ setting.
• Methodological Novelty: The significance lies in bypassing the standard Kloosterman sum bounds in the special case $h \approx N^2$ by exploiting the specific structure of the intervals via Ramanujan sums. This allows for a "square-root cancellation" error term that is not accessible through standard analytic methods applied to the general case.

The authors explicitly state that their results confirm the conjectured order of the error term ($O(N^{1+\varepsilon})$) in the specific regime $h = N^2 + O(N)$, a regime where previous techniques relying on divisibility properties of primes were insufficient.