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Keywords:
Davenport-Heilbronn method; prime varaible; exceptional set; Diophantine inequality
Davenport-Heilbronn method; prime varaible; exceptional set; Diophantine inequality
Summary:
Suppose that $\lambda _1,\lambda _2,\lambda _3,\lambda _4$ are nonzero real numbers, not all negative, $\delta > 0$, $\mathcal {V}$ is a well-spaced set, and the ratio $\lambda _1/\lambda _2$ is algebraic and irrational. Denote by $E(\mathcal {V}, N,\delta )$ the number of $v\in \mathcal {V}$ with $v\leq N$ such that the inequality $$ |\lambda _1p_1^2+\lambda _2p_2^3+\lambda _3p_3^4+\lambda _4p_4^5-v|<v^{-\delta } $$ has no solution in primes $p_1$, $p_2$, $p_3$, $p_4$. We show that $$ E(\mathcal {V}, N,\delta )\ll N^{1+2\delta -{1}/{72}+\varepsilon } $$ for any $\varepsilon >0$.
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