On the exceptional set for Diophantine inequality with unlike powers of primes

Abstract

Let λ₂, λ₃, λ₄, λ₅ be nonzero real numbers, not all negative. Let \(\mathfrak{V}\) be a well-spaced sequence. Assume that λ₂/λ₃ is irrational and algebraic, and δ > 0. Let \(E\left(\mathfrak{V},N,\delta \right)\) be the number of \(\upsilon \in \mathfrak{V}\) with \(\upsilon \le N\) such that the Diophantine inequality \(\left|{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{3}+{\lambda }_{4}{p}_{4}^{4}+{\lambda }_{5}{p}_{5}^{5}-\upsilon \right|<{\upsilon }^{-\delta }\) has no solution in primes p₂, p₃, p₄, p₅. In this paper, we prove that for any \(\varepsilon >0,E\left(\mathfrak{V},N,\delta \right)\ll {N}^{1-19/378+2\delta +\varepsilon },\) which refines the previous result.