Decoupling inequalities for short generalized Dirichlet sequences

TL;DR

This paper develops decoupling inequalities for functions whose Fourier transforms are supported near short Dirichlet sequences, using wave packet analysis to handle the arithmetic structure of these frequency sets.

Contribution

It introduces new decoupling inequalities tailored for short Dirichlet sequences and related convex frequency supports, expanding the scope of decoupling theory.

Findings

Established decoupling inequalities for functions near short Dirichlet sequences

Utilized wave packet analysis to handle arithmetic frequency structures

Extended decoupling methods to sequences with convexity properties

Abstract

We study decoupling theory for functions on $\mathbb{R}$ with Fourier transform supported in a neighborhood of short Dirichlet sequences $\{\log n\}_{n=N+1}^{N+N^{1/2}}$, as well as sequences with similar convexity properties. We utilize the wave packet structure of functions with frequency support near an arithmetic progression.