# Article

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Keywords:
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 $\varepsilon >0$.
References:
[1] Cook, R. J., Fox, A.: The values of ternary quadratic forms at prime arguments. Mathematika 48 (2001), 137-149. DOI 10.1112/S002557930001439X | MR 1996366 | Zbl 1035.11010
[2] Cook, R. J., Harman, G.: The values of additive forms at prime arguments. Rocky Mt. J. Math. 36 (2006), 1153-1164. DOI 10.1216/rmjm/1181069409 | MR 2274889 | Zbl 1140.11048
[3] Davenport, H.: Analytic Methods for Diophantine Equations and Diophantine Inequalities. The University of Michigan, Fall Semester 1962, Ann Arbor Publishers, Ann Arbor (1963). MR 0159786 | Zbl 1089.11500
[4] Ge, W., Li, W.: One Diophantine inequality with unlike powers of prime variables. J. Inequal. Appl. 2016 (2016), Paper No. 33, 8 pages. DOI 10.1186/s13660-016-0983-6 | MR 3453618 | Zbl 06547399
[5] Harman, G.: The values of ternary quadratic forms at prime arguments. Mathematika 51 (2004), 83-96. DOI 10.1112/S0025579300015527 | MR 2220213 | Zbl 1107.11043
[6] Harman, G.: Trigonometric sums over primes I. Mathematika 28 (1981), 249-254. DOI 10.1112/S0025579300010305 | MR 0645105 | Zbl 0465.10029
[7] Kumchev, A. V.: On Weyl sums over primes and almost primes. Mich. Math. J. 54 (2006), 243-268. DOI 10.1307/mmj/1156345592 | MR 2252758 | Zbl 1137.11054
[8] Languasco, A., Zaccagnini, A.: On a ternary Diophantine problem with mixed powers of primes. Acta Arith. 159 (2013), 345-362. DOI 10.4064/aa159-4-4 | MR 3080797 | Zbl 1330.11063
[9] Mu, Q., Lü, X. D.: Diophantine approximation with prime variables and mixed powers. Chin. Ann. Math., Ser. A 36 (2015), 303-312 Chinese. English summary. DOI 10.16205/j.cnki.cama.2015.0028 | MR 3443464 | Zbl 1340.11054
[10] Ren, X.: On exponential sums over primes and application in Waring-Goldbach problem. Sci. China Ser. A 48 (2005), 785-797. DOI 10.1360/03ys0341 | MR 2158973 | Zbl 1100.11025
[11] Schmidt, W. M.: Diophantine Approximation. Lecture Notes in Mathematics 785, Springer, New York (1980). DOI 10.1007/978-3-540-38645-2 | MR 0568710 | Zbl 0421.10019
[12] Vaughan, R. C.: Diophantine approximation by prime numbers I. Proc. Lond. Math. Soc., III. Ser. 28 (1974), 373-384. DOI 10.1112/plms/s3-28.2.373 | MR 0337812 | Zbl 0274.10045
[13] Vaughan, R. C.: Diophantine approximation by prime numbers II. Proc. Lond. Math. Soc., III. Ser. 28 (1974), 385-401. DOI 10.1112/plms/s3-28.3.385 | MR 0337813 | Zbl 0276.10031
[14] Yang, Y., Li, W.: One Diophantine inequality with integer and prime variables. J. Inequal. Appl. 2015 (2015), Paper No. 293, 9 pages. DOI 10.1186/s13660-015-0817-y | MR 3399256 | Zbl 1353.11065
[15] Zhao, L.: On the Waring-Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. (3) 108 (2014), 1593-1622. DOI 10.1112/plms/pdt072 | MR 3218320 | Zbl 06322154
[16] Zhao, L.: The additive problem with one cube and three cubes of primes. Mich. Math. J. 63 (2014), 763-779. DOI 10.1307/mmj/1417799225 | MR 3286670 | Zbl 136011092

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