Average Mahler’s measure and $L_p$ norms of Littlewood polynomials
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- by Stephen Choi and Tamás Erdélyi HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 1 (2014), 105-120
Abstract:
Littlewood polynomials are polynomials with each of their coefficients in the set $\{-1,1\}$. We compute asymptotic formulas for the arithmetic mean values of the Mahler’s measure and the $L_p$ norms of Littlewood polynomials of degree $n-1$. We show that the arithmetic means of the Mahler’s measure and the $L_p$ norms of Littlewood polynomials of degree $n-1$ are asymptotically $e^{-\gamma /2}\sqrt {n}$ and $\Gamma (1+p/2)^{1/p}\sqrt {n}$, respectively, as $n$ grows large. Here $\gamma$ is Euler’s constant. We also compute asymptotic formulas for the power means $M_{\alpha }$ of the $L_p$ norms of Littlewood polynomials of degree $n-1$ for any $p > 0$ and $\alpha > 0$. We are able to compute asymptotic formulas for the geometric means of the Mahler’s measure of the “truncated” Littlewood polynomials $\hat {f}$ defined by $\hat {f}(z) := \min \{|f(z)|,1/n\}$ associated with Littlewood polynomials $f$ of degree $n-1$. These “truncated” Littlewood polynomials have the same limiting distribution functions as the Littlewood polynomials. Analogous results for the unimodular polynomials, i.e., with complex coefficients of modulus $1$, were proved before. Our results for Littlewood polynomials were expected for a long time but looked beyond reach, as a result of Fielding known for means of unimodular polynomials was not available for means of Littlewood polynomials.References
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Additional Information
- Stephen Choi
- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
- MR Author ID: 319734
- Email: schoia@sfu.ca
- Tamás Erdélyi
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77842
- MR Author ID: 63810
- Email: terdelyi@math.tamu.edu
- Received by editor(s): January 30, 2014
- Received by editor(s) in revised form: April 15, 2014, and May 26, 2014
- Published electronically: October 29, 2014
- Additional Notes: The research of the first author was supported by NSERC of Canada
- Communicated by: Thomas Schlumprecht
- © Copyright 2014 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 1 (2014), 105-120
- MSC (2010): Primary 11C08, 30C10; Secondary 42A05, 60G99
- DOI: https://doi.org/10.1090/S2330-1511-2014-00013-4
- MathSciNet review: 3272724