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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extremal Mahler measures and $ L_s$ norms of polynomials related to Barker sequences


Authors: Peter Borwein, Stephen Choi and Jonas Jankauskas
Journal: Proc. Amer. Math. Soc. 141 (2013), 2653-2663
MSC (2010): Primary 11B83, 11C08, 30C10; Secondary 42A05, 94A05
Published electronically: April 26, 2013
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Abstract: In the present paper, we study the class $ \mathcal {L}P_n$ which consists of Laurent polynomials

$\displaystyle P(z) = (n+1) + \sum _{\substack {k = 1 \\ k \text { -- odd}}}^{n}c_k (z^k+z^{-k}), $

with all coefficients $ c_k$ equal to $ -1$ or $ 1$. Such polynomials arise in the study of Barker sequences of even length -- binary sequences with minimal possible autocorrelations. By using an elementary (but not trivial) analytic argument, we prove that polynomials $ R_n(z)$ with all coefficients $ c_k=1$ have minimal Mahler measures in the class $ \mathcal {L}P_n$. In conjunction with an estimate $ M(R_n)> n - 2/\pi \log {n} +O(1)$ proved in an earlier paper, we deduce that polynomials whose coefficients form a Barker sequence would possess unlikely large Mahler measures. A generalization of this result to $ L_s$ norms is also given.

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Additional Information

Peter Borwein
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada
Email: pborwein@sfu.ca

Stephen Choi
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada
Email: kkchoi@math.sfu.ca

Jonas Jankauskas
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
Email: jonas.jankauskas@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11545-5
PII: S 0002-9939(2013)11545-5
Keywords: Laurent polynomials, Barker conjecture, Barker polynomials, Barker sequences, Littlewood polynomials, Mahler measures, $L_p$ norms, aperiodic autocorrelations
Received by editor(s): August 8, 2011
Received by editor(s) in revised form: November 5, 2011, and November 10, 2011
Published electronically: April 26, 2013
Additional Notes: The first and second authors are supported by NSERC, Canada.
A visit of the third author at the IRMACS Center, Simon Fraser University, was funded by the Lithuanian Research Council (student research support project).
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.