Extremal Mahler measures and 𝐿_{𝑠} norms of polynomials related to Barker sequences

Borwein, Peter; Choi, Stephen; Jankauskas, Jonas

doi:10.1090/S0002-9939-2013-11545-5

Extremal Mahler measures and $L_s$ norms of polynomials related to Barker sequences
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Proc. Amer. Math. Soc. 141 (2013), 2653-2663 Request permission

Abstract:

In the present paper, we study the class $\mathcal {L}P_n$ which consists of Laurent polynomials \[ P(z) = (n+1) + \sum _{\substack {k = 1 \\ k \text { – odd}}}^{n}c_k (z^k+z^{-k}), \] 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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