Another application of Linnik's dispersion method

TL;DR

This paper extends Linnik's dispersion method to establish a new distribution estimate for narrow type-II sums involving divisor-bounded sequences, with implications for the Titchmarsh divisor problem.

Contribution

It provides a novel dispersion estimate for narrow type-II sums using Linnik's method and recent bounds on Kloosterman sums, advancing understanding of distribution in analytic number theory.

Findings

Achieved a distribution exponent of 1/2 + δ - ε for certain convolutions

Extended dispersion estimates to narrow type-II sums with divisor-bounded sequences

Applied results to the Titchmarsh divisor problem

Abstract

Let $\alpha_m$ and $\beta_n$ be two sequences of real numbers supported on $[M, 2M]$ and $[N, 2N]$ with $M = X^{1/2 - \delta}$ and $N = X^{1/2 + \delta}$. We show that there exists a $\delta_0 > 0$ such that the multiplicative convolution of $\alpha_m$ and $\beta_n$ has exponent of distribution $\frac{1}{2} + \delta-\varepsilon$ (in a weak sense) as long as $0 \leq \delta < \delta_0$, the sequence $\beta_n$ is Siegel-Walfisz and both sequences $\alpha_m$ and $\beta_n$ are bounded above by divisor functions. Our result is thus a general dispersion estimate for "narrow" type-II sums. The proof relies crucially on Linnik's dispersion method and recent bounds for trilinear forms in Kloosterman fractions due to Bettin-Chandee. We highlight an application related to the Titchmarsh divisor problem.