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

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Abstract

Suppose that λ1, λ2, λ3, λ4 are nonzero real numbers, not all negative, δ > 0, V is a well-spaced set, and the ratio λ12 is algebraic and irrational. Denote by E(V,N, δ) the number of vV with vN such that the inequality

$$\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^3 + {\lambda _3}p_3^4 + {\lambda _4}p_4^5 - \upsilon } \right| < {\upsilon ^{ - \delta }}$$

has no solution in primes p1, p2, p3, p4. We show that

$$E\left( {\upsilon ,N,\delta } \right) \ll {N^{1 + 2\delta - 1/72 + \varepsilon }}$$

for any ɛ > 0.

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Correspondence to Wenxu Ge.

Additional information

The research has been supported by the National Natural Science Foundation of China (Grants No. 11371122, 11471112), and the Research Foundation of North China University of Water Resources and Electric Power (No. 201084).

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Ge, W., Zhao, F. The exceptional set for Diophantine inequality with unlike powers of prime variables. Czech Math J 68, 149–168 (2018). https://doi.org/10.21136/CMJ.2018.0388-16

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  • DOI: https://doi.org/10.21136/CMJ.2018.0388-16

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