Abstract
In this paper we study the two-dimensional analog of the classical quadratic Weyl sum. We establish a major arc approximation with a strong error term akin to celebrated results of R.C. Vaughan from the early 1980s and the late 2000s. Our main result has applications to the study of discrete maximal operators related to triangular configurations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
T. C. Anderson, A. V. Kumchev, and E. A. Palsson, Discrete maximal operators for surfaces of higher codimension, preprint arXiv:2006.09968.
G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba, Trigonometric Sums in Number Theory and Analysis, Walter de Gruyter GmbH & Co., Berlin, 2004.
J. Brandes, S. T. Parsell, k. Poulias, G. Shakan, and R. C. Vaughan, On generating functions in additive number theory, II: Lower-order terms and applications to PDEs, Math. Ann. 379 (2021), 347–376.
J. Brüdern and O. Robert, Rational points on linear slices of diagonal hypersurfaces, Nagoya Math. J. 218 (2015), 51–100.
R. C. Vaughan, Some remarks on Weyl sums, in “Topics in Classical Number Theory,” Vol. I, II (Budapest, 1981), North-Holland, Amsterdam, 1984, pp. 1585–1602.
R. C. Vaughan, The Hardy–Littlewood Method, Second ed., Cambridge University Press, Cambridge, 1997.
R. C. Vaughan, On generating functions in additive number theory I, in “Analytic Number Theory. Essays in Honour of Klaus Roth”, Cambridge University Press, Cambridge, 2009, pp. 436–448.
H. Weyl, Über die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77 (1916), no. 3, 313–352.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Kumchev, A.V. (2021). On a Two-Dimensional Exponential Sum. In: Nathanson, M.B. (eds) Combinatorial and Additive Number Theory IV. CANT 2020. Springer Proceedings in Mathematics & Statistics, vol 347. Springer, Cham. https://doi.org/10.1007/978-3-030-67996-5_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-67996-5_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-67995-8
Online ISBN: 978-3-030-67996-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)