Extremal Mahler measures and $L_s$ norms of polynomials related to Barker sequences
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Abstract:
In the present paper, we study the class $\mathcal {L}P_n$ which consists of Laurent polynomials \[ P(z) = (n+1) + \sum _{\substack {k = 1 \\ k \text { â odd}}}^{n}c_k (z^k+z^{-k}), \] with all coefficients $c_k$ equal to $-1$ or $1$. Such polynomials arise in the study of Barker sequences of even length â 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.References
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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
- MR Author ID: 825362
- ORCID: 0000-0001-9770-7632
- Email: jonas.jankauskas@gmail.com
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2653-2663
- MSC (2010): Primary 11B83, 11C08, 30C10; Secondary 42A05, 94A05
- DOI: https://doi.org/10.1090/S0002-9939-2013-11545-5
- MathSciNet review: 3056555