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BOUNDS FOR TRIPLE EXPONENTIAL SUMS WITH MIXED EXPONENTIAL AND LINEAR TERMS

Part of: Exponential sums and character sums Diophantine equations

Published online by Cambridge University Press:  03 May 2018

KAM HUNG YAU
KAM HUNG YAU*
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
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Abstract

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We establish bounds for triple exponential sums with mixed exponential and linear terms. The method we use is by Shparlinski [‘Bilinear forms with Kloosterman and Gauss sums’, Preprint, 2016, arXiv:1608.06160] together with a bound for the additive energy from Roche-Newton et al. [‘New sum-product type estimates over finite fields’, Adv. Math.293 (2016), 589–605].

Keywords

exponential sumsexponential functioncancellationsolutions to congruences in small boxes

MSC classification

Primary: 11L07: Estimates on exponential sums
Secondary: 11D79: Congruences in many variables
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 1 , August 2018 , pp. 64 - 69
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Kerr, B., ‘Incomplete exponential sums over exponential functions’, Q. J. Math. 66(1) (2015), 213–224.Google Scholar
Konyagin, S. V. and Shparlinski, I. E., Character Sums with Exponential Functions and their Applications, Cambridge Tracts in Mathematics, 136 (Cambridge University Press, Cambridge, 1999).Google Scholar
Roche-Newton, O., Rudnev, M. and Shkredov, I. D., ‘New sum-product type estimates over finite fields’, Adv. Math. 293 (2016), 589–605.Google Scholar
Shparlinski, I. E., Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness, Progress in Computer Science and Applied Logic, 22 (Birkhäuser, Basel, 2013).Google Scholar
Shparlinski, I. E., ‘Bilinear forms with Kloosterman and Gauss sums’, Preprint, 2016, arXiv:1608.06160.Google Scholar
Shparlinski, I. E. and Yau, K. H., ‘Bounds of double multiplicative character sums and gaps between residues of exponential functions’, J. Number Theory 167 (2016), 304–316.Google Scholar
Shparlinski, I. E. and Yau, K. H., ‘Double exponential sums with exponential functions’, Int. J. Number Theory 13 (2017), 2531–2543.Google Scholar
Vinogradov, I. M., The Method of Trigonometrical Sums in the Theory of Numbers (Interscience Publishers, New York, NY, 1954).Google Scholar