The exceptional set for Diophantine inequality with mixed powers of primes

TL;DR

This paper investigates the size of the exceptional set of well-spaced numbers for which a Diophantine inequality involving mixed powers of primes has no solutions, providing upper bounds under certain conditions.

Contribution

It establishes upper bounds on the exceptional set for a Diophantine inequality with mixed prime powers, extending previous results to more general cases.

Findings

Upper bounds on the exceptional set size for the inequality

Results hold for k ≥ 5 and any ε > 0

The inequality involves mixed powers of primes with irrational ratios

Abstract

Assume that $\lambda_1, \lambda_2, \lambda_3,\lambda_4,\lambda_5,\lambda_6,\lambda_7$ are non-zero real numbers , $\lambda_1/\lambda_2$ is an irrational number. Let $\mathcal{V} $ be a well-spaced sequence, and $\delta >0$. For any given positive integer $k\geq 5$ and any $\varepsilon >0$, we give the upper bound of the number of $\upsilon \in \mathcal{V} $ with $\upsilon \leq X$ for which the inequality $$ \left | \lambda_1p_1^2 + \lambda_2p_2^3 + \lambda_3p_3^3 + \lambda_4p_4^3 + \lambda_5p_5^3 + \lambda_6p_6^4 + \lambda_7p_7^k - \upsilon \right | <{\upsilon}^{-\delta} $$ has no solution in primes $p_1, p_2, p_3, p_4, p_5, p_6, p_7$.