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The correlation measures of finite sequences: limiting distributions and minimum values


Author: Kai-Uwe Schmidt
Journal: Trans. Amer. Math. Soc. 369 (2017), 429-446
MSC (2010): Primary 11K45; Secondary 60C05, 68R15
DOI: https://doi.org/10.1090/tran6650
Published electronically: March 21, 2016
MathSciNet review: 3557779
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Abstract: Three measures of pseudorandomness of finite binary sequences were introduced by Mauduit and Sárközy in 1997 and have been studied extensively since then: the normality measure, the well-distribution measure, and the correlation measure of order $ r$. Our main result is that the correlation measure of order $ r$ for random binary sequences converges strongly, and so has a limiting distribution. This solves a problem due to Alon, Kohayakawa, Mauduit, Moreira, and Rödl. We also show that the best known lower bounds for the minimum values of the correlation measures are simple consequences of a celebrated result due to Welch concerning the maximum nontrivial scalar products over a set of vectors.


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Additional Information

Kai-Uwe Schmidt
Affiliation: Faculty of Mathematics, Otto-von-Guericke University, Universitätsplatz 2, 39106 Magdeburg, Germany
Address at time of publication: Department of Mathematics, Paderborn University, Warburger Strasse 100, 33098 Paderborn, Germany
Email: kaiuwe.schmidt@ovgu.de, kus@math.upb.de

DOI: https://doi.org/10.1090/tran6650
Received by editor(s): January 10, 2014
Received by editor(s) in revised form: November 24, 2014, and January 6, 2015
Published electronically: March 21, 2016
Article copyright: © Copyright 2016 American Mathematical Society