Diophantine approximation with a quaternary problem

TL;DR

This paper proves that a specific Diophantine inequality involving prime variables and quadratic and fractional powers has infinitely many solutions under certain conditions, advancing understanding in prime-based Diophantine approximation.

Contribution

It establishes the existence of infinitely many solutions to a new class of prime-based Diophantine inequalities with mixed powers, extending previous results.

Findings

Infinitely many solutions exist for the inequality under given conditions.

The result applies to primes with mixed quadratic and fractional powers.

The approximation error bound is explicitly characterized.

Abstract

Let $1<k<7/6$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign such that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. We prove that the inequality $|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^k-\omega|\le (\max_j p_j)^{-\frac{7-6k}{14k}+\varepsilon}$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\varepsilon>0$.