Taiwanese Journal of Mathematics

Diophantine Approximation with Mixed Powers of Primes

Huafeng Liu and Jing Huang

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

Let $k$ be an integer with $k \geq 3$. Let $\lambda_1$, $\lambda_2$, $\lambda_3$ be non-zero real numbers, not all negative. Assume that $\lambda_1/\lambda_2$ is irrational and algebraic. Let $\mathcal{V}$ be a well-spaced sequence, and $\delta \gt 0$. In this paper, we prove that, for any $\varepsilon \gt 0$, the number of $\upsilon \in \mathcal{V}$ with $\upsilon \leq X$ such that the inequality \[ |\lambda_1 p_1^2 + \lambda_2 p_2^2 + \lambda_3 p_3^k - \upsilon| \lt \upsilon^{-\delta} \] has no solution in primes $p_1$, $p_2$, $p_3$ does not exceed $O(X^{1-2/(7m_2(k))+2\delta+\varepsilon})$, where $m_2(k)$ relies on $k$. This refines a recent result. Furthermore, we briefly describe how a similar method can refine a previous result on a Diophantine problem with two squares of primes, one cube of primes and one $k$-th power of primes.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 18 pages.

Dates
First available in Project Euclid: 23 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1550890836

Digital Object Identifier
doi:10.11650/tjm/190201

Subjects
Primary: 11P32: Goldbach-type theorems; other additive questions involving primes 11P55: Applications of the Hardy-Littlewood method [See also 11D85] 11N36: Applications of sieve methods

Keywords
Waring-Goldbach problem Diophantine inequality Sieve methods

Citation

Liu, Huafeng; Huang, Jing. Diophantine Approximation with Mixed Powers of Primes. Taiwanese J. Math., advance publication, 23 February 2019. doi:10.11650/tjm/190201. https://projecteuclid.org/euclid.twjm/1550890836


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