The exceptional set for Diophantine approximation with mixed powers of prime variables

2023/12/11 by Yuhui Liu, Liu, Yuhui
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2312.06110

openalex publication_date 2023/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let lambda₁, λ₂, λ₃, λ₄ be non-zero real numbers, not all negative, with λ₁/λ₂ irrational and algebraic. Suppose that V is a well-spaced sequence and δ>0. In this paper, it is proved that for any ε >0, the number of v ∈ V with v \leqslant N for which |λ₁ p₁² + λ₂ p₂³+ λ₃ p₃⁴+ λ₄ p₄⁵ - v| < v^-δ has no solution in prime variables p₁,p₂,p₃,p₄ does not exceed O(N^{(359)/(378) + 2δ+ε}). This result constitutes an improvement upon that of Q. W. Mu and Z. P. Gao [12].

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