Czechoslovak Mathematical Journal, Vol. 68, No. 1, pp. 149-168, 2018
The exceptional set for Diophantine inequality with unlike powers of prime variables
Wenxu Ge, Feng Zhao
Received July 22, 2016. First published January 17, 2018.
Abstract: 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 $\varepsilon>0$.
Keywords: Davenport-Heilbronn method; prime varaible; exceptional set; Diophantine inequality
Affiliations: Wenxu Ge (corresponding author), Feng Zhao, Department of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Jinshui E Rd, Jinshui, Zhengzhou, 450046, Henan, P. R. China, e-mail: gewenxu1982@163.com, gewenxu@ncwu.edu.cn, zhaofeng@ncwu.edu.cn
