Average Mahler’s measure and $L_p$ norms of Littlewood polynomials

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.