Another application of Linnik's dispersion method

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 $\frac{1}{2}+\delta -\epsilon$ (in a weak sense) as long as $0\le \delta <{\delta }_0$, 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.

19 pages