# On the ergodic Waring--Goldbach problem

## TL;DR

This paper establishes an asymptotic formula for the Fourier transform related to the Waring--Goldbach problem and applies it to bounds on discrete spherical maximal functions along primes and ergodic distribution results.

## Contribution

It introduces a new asymptotic analysis of the Fourier transform of the arithmetic surface measure in the Waring--Goldbach problem, leading to several applications.

## Key findings

- Derived an asymptotic formula for the Fourier transform of the arithmetic surface measure.
- Provided bounds for discrete spherical maximal functions along primes.
- Established ergodic theorems related to the distribution of primes.

## Abstract

We prove an asymptotic formula for the Fourier transform of the arithmetic surface measure associated to the Waring--Goldbach problem and provide several applications, including bounds for discrete spherical maximal functions along the primes and distribution results such as ergodic theorems.