Upper Bounds for Double Exponential Sums Along a Subsequence

Christopher J. White

doi:10.1515/udt-2017-0012

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Upper Bounds for Double Exponential Sums Along a Subsequence

Uniform distribution theory

Volume 12 (2017): Issue 2 (December 2017)

By: Christopher J. White

Open Access

|Jan 2018

Abstract

We consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai. They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of α, provided α is badly-approximable. We provide a proof of a result, which includes a simple proof of their theorem, and which applies for all irrational α.

References

[1] BERESNEVICH, V.-HAYNES, A.-VELANI, S.: Sums of reciprocals of (nα−γ) modulo 1 and multiplicative Diophantine approximation, preprint.
Search in Google Scholar Back to article
[2] COSENTINO, S.-FLAMINIO, L.: Equidistribution for higher-rank abelian actions on Heisenberg nilmanifolds, J. Mod. Dyn. 9 (2015), 305-353.10.3934/jmd.2015.9.305
Search in Google Scholar Back to article
[3] HARDY, G. H.-LITTLEWOOD, J. E.: Some problems of Diophantine approximation (a series of cosecants), Bull. Calcutta Math. Soc. 20 (1930), 251-266.
Search in Google Scholar Back to article
[4] HARDY, G. H-LITTLEWOOD, J. E.-P´OLYA, G.: Inequalities, Cambridge Mathematical Library (2nd edition), Cambridge University Press, Cambridge, 1952.
Search in Google Scholar Back to article
[5] HARMAN, G.: Metric Number Theory, In: London Mathematical Society Monographs. New Series, Vol. 18. The Clarendon Press, Oxford University Press, New York, 1998.
Search in Google Scholar Back to article
[6] HAYNES, A.: Equivalence classes of codimension one cut-and-project sets, Ergodic Theory Dyn. Syst. 36 (2016), no. 3, 816-831.
Search in Google Scholar Back to article
[7] KHINCHIN, A.YA.: Continued Fractions, The University of Chicago Press, Chicago, III.-London, 1964.10.1063/1.3051235
Search in Google Scholar Back to article
[8] ROCKETT, A.-SZ¨USZ, P.: Continued Fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992.10.1142/1725
Search in Google Scholar Back to article
[9] SINAI, YA.G.-ULCGRAI, C.: Estimates from above of certain double trigonometric sums, J. Fixed Point Theory Appl. 6 (2009), no. 1, 93-113.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.1515/udt-2017-0012 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 11 - 24

Submitted on: Nov 1, 2015

Accepted on: Oct 10, 2016

Published on: Jan 30, 2018

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

11J70,

11L03,

11L07

Related subjects:

Mathematics,

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2018 Christopher J. White, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 12 (2017): Issue 2 (December 2017)