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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The correlation measures of finite sequences: limiting distributions and minimum values
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by Kai-Uwe Schmidt PDF
Trans. Amer. Math. Soc. 369 (2017), 429-446 Request permission

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.
References
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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
  • MR Author ID: 789692
  • Email: kaiuwe.schmidt@ovgu.de, kus@math.upb.de
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 429-446
  • MSC (2010): Primary 11K45; Secondary 60C05, 68R15
  • DOI: https://doi.org/10.1090/tran6650
  • MathSciNet review: 3557779