# A Diophantine approximation problem with two primes and one $k$-th power of a prime

## TL;DR

This paper improves bounds on a Diophantine approximation problem involving two primes and a prime's k-th power, extending the range of k and providing stronger approximation bounds.

## Contribution

The authors extend the range of k from 1<k<4/3 to 1<k≤3 and enhance approximation bounds using harmonic analysis techniques.

## Key findings

- Extended the k-range to 1<k≤3
- Combined Harman's technique with new estimates for exponential sums
- Provided stronger bounds for Diophantine approximation

## Abstract

We refine a result of the last two Authors of [8] on a Diophantine approximation problem with two primes and a $k$-th power of a prime which was only proved to hold for $1<k<4/3$. We improve the $k$-range to $1<k\le 3$ by combining Harman's technique on the minor arc with a suitable estimate for the $L^4$-norm of the relevant exponential sum over primes $S_k$. In the common range we also give a stronger bound for the approximation.