A Diophantine approximation problem with unlike powers of primes

Xinyan Li; Wenxu Ge; Xinyan Li; Wenxu Ge

doi:10.3934/math.2025034

AIMS Mathematics

2025, Volume 10, Issue 1: 736-753. doi: 10.3934/math.2025034

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A Diophantine approximation problem with unlike powers of primes

Xinyan Li ^1,2,
Wenxu Ge ^{1
,
,}

1.
School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, 450046, China
2.
Institute of Mathematics, Henan Academy of Sciences, Zhengzhou, 450046, China

Received: 04 November 2024 Revised: 17 December 2024 Accepted: 25 December 2024 Published: 13 January 2025
MSC : 11D75, 11P32, 11P55

Let $ \lambda_{1} $, $ \lambda_{2} $, $ \lambda_{3} $, and $ \lambda_{4} $ be non-zero real numbers, not all negative. Suppose that $ {{{\lambda }_{1}}}/{{{\lambda }_{3}}}\; $is irrational and algebraic, $ \delta > 0 $, and the set $ \mathcal{V} $ is a well-spaced sequence. In this paper, we prove that, for any $ \varepsilon > 0 $, the number of $ v\in \mathcal{V} $ with $ v\le X $ such that the inequality
$ \begin{align} \left| {{\lambda }_{1}}{{p}_{1}}+{{\lambda }_{2}}{{p}_{2}}^{2}+{{\lambda }_{3}}{{p}_{3}}^{3}+{{\lambda }_{4}}{{p}_{4}}^{4}-v \right|<{{v}^{-\delta }} \end{align} $
has no solution in primes $ {{p}_{1}} $, $ {{p}_{2}} $, $ {{p}_{3}} $, $ {{p}_{4}} $ that does not exceed $ O({{X}^{1-\frac{83}{144}+2\delta +2\varepsilon }}) $.
- Davenport-Heilbronn method,
- Diophantine inequality,
- primes,
- exceptional set
Citation: Xinyan Li, Wenxu Ge. A Diophantine approximation problem with unlike powers of primes[J]. AIMS Mathematics, 2025, 10(1): 736-753. doi: 10.3934/math.2025034

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Abstract

Let $ \lambda_{1} $, $ \lambda_{2} $, $ \lambda_{3} $, and $ \lambda_{4} $ be non-zero real numbers, not all negative. Suppose that $ {{{\lambda }_{1}}}/{{{\lambda }_{3}}}\; $is irrational and algebraic, $ \delta > 0 $, and the set $ \mathcal{V} $ is a well-spaced sequence. In this paper, we prove that, for any $ \varepsilon > 0 $, the number of $ v\in \mathcal{V} $ with $ v\le X $ such that the inequality

$ \begin{align} \left| {{\lambda }_{1}}{{p}_{1}}+{{\lambda }_{2}}{{p}_{2}}^{2}+{{\lambda }_{3}}{{p}_{3}}^{3}+{{\lambda }_{4}}{{p}_{4}}^{4}-v \right|<{{v}^{-\delta }} \end{align} $

has no solution in primes $ {{p}_{1}} $, $ {{p}_{2}} $, $ {{p}_{3}} $, $ {{p}_{4}} $ that does not exceed $ O({{X}^{1-\frac{83}{144}+2\delta +2\varepsilon }}) $.

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