Czechoslovak Mathematical Journal, first online, pp. 1-20


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
Classification MSC:  11D75, 11P32, 11P55
DOI:  10.21136/CMJ.2018.0388-16

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] R. J. Cook, A. Fox: The values of ternary quadratic forms at prime arguments. Mathematika 48 (2001), 137-149. DOI 10.1112/S002557930001439X | MR 1996366 | Zbl 1035.11010
[2] R. J. Cook, G. Harman: 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] H. Davenport: 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] W. Ge, W. Li: 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] G. Harman: The values of ternary quadratic forms at prime arguments. Mathematika 51 (2004), 83-96. DOI 10.1112/S0025579300015527 | MR 2220213 | Zbl 1107.11043
[6] G. Harman: Trigonometric sums over primes I. Mathematika 28 (1981), 249-254. DOI 10.1112/S0025579300010305 | MR 0645105 | Zbl 0465.10029
[7] A. V. Kumchev: 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] A. Languasco, A. Zaccagnini: 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] Q. Mu, X. D. Lü: Diophantine approximation with prime variables and mixed powers. Chin. Ann. Math., Ser. A 36 (2015), 303-312. (In Chinese. English summary.) DOI 10.16205/j.cnki.cama.2015.0028 | MR 3443464 | Zbl 1340.11054
[10] X. Ren: 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] W. M. Schmidt: 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] R. C. Vaughan: 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] R. C. Vaughan: 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] Y. Yang, W. Li: 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] L. Zhao: 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] L. Zhao: 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

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


 
PDF available at: