Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions

1. Problem Statement

The paper addresses a central problem in analytic number theory concerning the distribution of arithmetic functions in arithmetic progressions, specifically focusing on unbalanced convolutions.

Given two sequences of complex numbers, $\alpha_m$ (supported on $m \sim M$) and $\beta_n$ (supported on $n \sim N$), the goal is to estimate the error term in the distribution of their product $mn$ modulo $q$. Specifically, the author investigates the quantity:
$$\Delta(\alpha, \beta, M, N, Q, a) = \sum_{q \sim Q} \left| \sum_{\substack{mn \equiv a \pmod q \\ m \sim M, n \sim N}} \alpha_m \beta_n - \frac{1}{\phi(q)} \sum_{\substack{mn \equiv a \pmod q \\ (mn,q)=1}} \alpha_m \beta_n \right|$$
The objective is to establish bounds of the form $\ll X (\log X)^{-A}$ for moduli $Q$ up to $X^{1/2 + \delta}$, where $X \approx MN$.

The Challenge:
While the generalized Bombieri-Vinogradov theorem handles cases where $\delta < 0$, proving results for $\delta > 0$ (i.e., extending the range of moduli beyond the square root of the conductor) is significantly harder. Previous work by Fouvry and Radziwiłł (2018) established such bounds for small $\delta > 0$, but only under restrictive conditions where $N$ (the length of the shorter sequence) was very small relative to $X$ (specifically $N \le X^{1/72}$). The paper aims to relax these constraints on $N$ and $Q$.

2. Methodology

The core innovation of the paper lies in refining the estimation of trilinear forms involving Kloosterman fractions.

2.1. The Bettin-Chandee Framework

The methodology builds upon a famous result by Bettin and Chandee (2011) regarding trilinear forms with Kloosterman fractions. The standard form analyzed is:
$$B(M, N, A) = \sum_{a \sim A, m \sim M, n \sim N} \alpha_m \beta_n \nu_a e\left( \frac{\vartheta am}{n} \right)$$
Fouvry and Radziwiłł applied this theorem to their problem by effectively treating a fixed factor in the denominator as part of the variable $n$, which resulted in a loss of efficiency (replacing $N$ with $NQ$ in the bounds).

2.2. The "Partially Fixed Moduli" Improvement

Wright introduces a novel variation where the denominator contains a fixed factor $R$. He defines a modified trilinear form:
$$B(M, N, A; R) = \sum_{a \in A, m \in M, n \in N} \alpha_m \beta_n \nu_a e\left( \frac{\vartheta am}{nR} \right)$$
where $R$ is a parameter such that $R \ll M A$.

Key Technical Steps:

1. Complementary Divisors: Wright adapts the "complementary divisor" mechanism used by Bettin and Chandee. Instead of introducing an auxiliary variable $b$ to handle the denominator, he utilizes the existing fixed factor $R$ in the denominator.
2. Refined Bounds: By analyzing the sum with the fixed factor $R$ explicitly, he derives a new bound for $B(M, N, A; R)$ that is superior to simply substituting $N \to NR$ in the original Bettin-Chandee theorem.
3. Dispersion Method: The improved bound on the trilinear form is then fed into the dispersion method (specifically the Linnik dispersion technique used by Fouvry and Radziwiłł) to bound the error term $\Delta$. This involves decomposing the error into main terms and error terms, where the new trilinear bound specifically improves the estimation of the error term $\tilde{W}_{Err1}$.

3. Key Contributions

The paper makes two primary technical contributions:

1. Theorem 2.1 (Improved Trilinear Bound):
   Wright proves a new bound for trilinear forms with a fixed factor $R$ in the denominator.
   $$B(M, N, A; R) \ll M^\epsilon \|\alpha\| \|\nu\| \|\beta\| (AMN)^{1/2} R^{1/4} \left( 1 + \frac{|\vartheta|A}{MN} \right)^{1/4} \times (\text{polynomial terms in } M, N, A, R)$$
   Crucially, when $R$ is large, this bound is significantly tighter than the naive application of the original Bettin-Chandee theorem.

2. Theorem 2.3 (Improved Dispersion Bound):
   By applying Theorem 2.1 to the dispersion method, Wright derives a sharper bound for the dispersion $\Delta$.

   • Previous Result (Fouvry-Radziwiłł): The error term contained components like $Q^{15/8} N^{23/8}$.
   • New Result: The error term is reduced to $Q^{15/8} N^{11/4}$.
   • Savings: This represents a saving of a factor of $N^{1/8}$ in the first term and $N^{2/5}$ in the second term of the error estimate.

4. Main Results

The improved error bounds allow for significantly wider ranges of validity for the unbalanced convolution estimates (Corollary 2.2).

Corollary 2.2 establishes that the desired bound $\ll X (\log X)^{-A}$ holds under the following conditions (where $MN \approx X$):

• Case (i) - Improved Range for $N$:
  If $Q \le X^{1/2 + 1/66 - \delta}$, the allowable range for $N$ is improved from $N \le X^{1/72}$ (previous) to:
  $$N \le Q^{-33/28} X^{17/28 - \epsilon}$$
  Specifically, if $Q = X^{1/2 + \epsilon}$, this allows $N \le X^{1/56 - \epsilon'}$, a substantial improvement over $X^{1/72}$.

• Case (ii) & (iii) - Improved Range for $Q$:
  For fixed $N$ in the range $X^{1/89} \le N \le X^{7/90}$, the allowable modulus $Q$ is improved from $X^{53/105}$ to:
  $$Q \le X^{45/89 - \epsilon}$$
  (Note: $45/89 \approx 0.5056$, whereas $53/105 \approx 0.5047$).

Additionally, Corollary 2.4 extends these improvements to the distribution of divisor sums (related to the divisor function), improving the range of moduli $Q$ for which the Bombieri-Vinogradov theorem holds in this context.

5. Significance

• Breaking Barriers: The paper pushes the boundaries of the "unbalanced" case in the Bombieri-Vinogradov theorem. In analytic number theory, the ability to handle larger moduli $Q$ (closer to $X^{1/2}$) or longer sequences $N$ is critical for problems involving primes, such as the Elliott-Halberstam conjecture.
• Methodological Refinement: The work demonstrates that the "fixed factor" in the denominator of Kloosterman sums, often treated as a nuisance or absorbed into variables, can be exploited structurally to gain savings. This suggests a new avenue for refining exponential sum estimates.
• Foundation for Future Work: As the first part of a series ("Part I"), this paper lays the groundwork for further improvements. The techniques developed here are likely to be applied to more complex convolution problems or to push the exponent $\delta$ closer to the conjectured limits.

In summary, Thomas Wright successfully refines the analysis of trilinear Kloosterman fractions by exploiting partially fixed moduli, resulting in tighter error bounds that expand the known range of validity for unbalanced convolution estimates in analytic number theory.