Taiwanese Journal of Mathematics

Diophantine Approximation with Mixed Powers of Primes

Huafeng Liu and Jing Huang

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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

Taiwanese J. Math., Volume 23, Number 5 (2019), 1073-1090.

Received: 22 November 2018
Revised: 30 December 2018
Accepted: 13 February 2019
First available in Project Euclid: 23 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Waring-Goldbach problem Diophantine inequality Sieve methods


Liu, Huafeng; Huang, Jing. Diophantine Approximation with Mixed Powers of Primes. Taiwanese J. Math. 23 (2019), no. 5, 1073--1090. doi:10.11650/tjm/190201. https://projecteuclid.org/euclid.twjm/1550890836

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