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Distribution of short subsequences of inversive congruential pseudorandom numbers modulo $ 2^t$


Authors: László Mérai and Igor E. Shparlinski
Journal: Math. Comp. 89 (2020), 911-922
MSC (2010): Primary 11K38, 11K45, 11L07
DOI: https://doi.org/10.1090/mcom/3467
Published electronically: September 9, 2019
MathSciNet review: 4044455
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Abstract: In this paper we study the distribution of very short sequences of inversive congruential pseudorandom numbers modulo $ 2^t$. We derive a new bound on exponential sums with such sequences and use it to estimate their discrepancy. The technique we use is based on the method of N. M. Korobov (1972) of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford (2002), which has never been used in this area and is very likely to find further applications.


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Additional Information

László Mérai
Affiliation: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria
Email: laszlo.merai@oeaw.ac.at

Igor E. Shparlinski
Affiliation: School of Mathematics and Statistics, University of New South Wales. Sydney, New South Wales 2052, Australia
Email: igor.shparlinski@unsw.edu.au

DOI: https://doi.org/10.1090/mcom/3467
Keywords: Inversive congruential pseudorandom numbers, prime powers, exponential sums, Vinogradov mean value theorem
Received by editor(s): December 19, 2018
Received by editor(s) in revised form: April 29, 2019, and May 9, 2019
Published electronically: September 9, 2019
Additional Notes: During the preparation of this work the first author was partially supported by the Austrian Science Fund FWF Projects P30405 and the second author by the Australian Research Council Grants DP170100786 and DP180100201.
Article copyright: © Copyright 2019 American Mathematical Society