The exceptional set for Diophantine inequality with unlike powers of prime variables

Ge, Wenxu, and Zhao, Feng. "The exceptional set for Diophantine inequality with unlike powers of prime variables." Czechoslovak Mathematical Journal 68.1 (2018): 149-168. <http://eudml.org/doc/294289>.

@article{Ge2018,
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 $v\le N$ 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 $\varepsilon >0$.},
author = {Ge, Wenxu, Zhao, Feng},
journal = {Czechoslovak Mathematical Journal},
keywords = {Davenport-Heilbronn method; prime varaible; exceptional set; Diophantine inequality},
language = {eng},
number = {1},
pages = {149-168},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The exceptional set for Diophantine inequality with unlike powers of prime variables},
url = {http://eudml.org/doc/294289},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Ge, Wenxu
AU - Zhao, Feng
TI - The exceptional set for Diophantine inequality with unlike powers of prime variables
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 1
SP - 149
EP - 168
AB - 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 $v\le N$ 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 $\varepsilon >0$.
LA - eng
KW - Davenport-Heilbronn method; prime varaible; exceptional set; Diophantine inequality
UR - http://eudml.org/doc/294289
ER -