Mahler's problem and Turyn polynomials

TL;DR

The paper develops a probabilistic framework to analyze the asymptotic behavior of the Mahler measure and $L_q$ norms for Turyn polynomials $F_{p,t}$, which shift Fekete polynomial coefficients by a prescribed amount. By modeling the critical interpolants with a random process $G_{\mathbb{X},\alpha}(x)$ and proving convergence in law, the authors derive explicit limiting values $\kappa_q(\alpha)$ for all $q>0$ and $\alpha\in[0,1]$, and $\kappa_0(\alpha)$ for the Mahler measure. For the notable shift $\alpha=1/4$, they obtain a record asymptotic normalized Mahler measure $M(F_{p,\lfloor p/4\rfloor})/\sqrt{p} \to 0.951...$, and they show companion Littlewood polynomials preserve these limits, addressing Newman-type questions. Numerical calculations reinforce the theoretical results, yielding $\kappa_0(0)\approx0.740$, $\kappa_0(1/4)\approx0.951$, and $\kappa_1(1/4)\approx0.978$, $\kappa_3(1/4)\approx1.020$, with implications for optimal merit factors and $L_1$ norms. The work also connects to generalized Turyn polynomials, suggesting avenues for even larger Mahler measures and further interplay between random models and analytic properties of sign polynomials.

Abstract

Mahler's problem asks for the largest possible value of the Mahler measure, normalized by the $L_2$ norm, of a polynomial with $\pm1$ coefficients and large degree. We establish a new record value in this problem exceeding $0.95$ by analyzing certain Turyn polynomials, which are defined by cyclically shifting the coefficients of a Fekete polynomial by a prescribed amount. It was recently established that the distribution of values over the unit circle of Fekete polynomials of large degree is effectively modeled by a particular random point process. We extend this analysis to the Turyn polynomials, and determine expressions for the asymptotic normalized Mahler measure of these polynomials, as well as for their normalized $L_q$ norms. We also describe a number of calculations on the corresponding random processes, which indicate that the Turyn polynomials where the shift is approximately $1/4$ of the length have Mahler measure exceeding $95\%$ of their $L_2$ norm. Further, we show that these asymptotic values are not disturbed by a small change to make polynomials having entirely $\pm1$ coefficients, which establishes the result on Mahler's problem. We also estimate that the limiting value of the normalized $L_1$ norm of these polynomials exceeds $0.977$, in connection with a question of Newman.

Figures (5)

• Figure 6.1: $\kappa_0^5(\alpha)$ (lowest), $\kappa_0^8(\alpha)$, and $\kappa_0^{14}(\alpha)$ (highest) for $0\leq\alpha\leq1/4$.
• Figure 6.2: $\kappa_0^{14}(\alpha)-\kappa_0^{13}(\alpha)$ for $0\leq\alpha\leq 1/4$.
• Figure 6.3: $\kappa_0^J(0)$ for $J\leq18$ (solid points), their interpolating curve and its asymptotic value (solid curve and dashed line), and sampling estimates for $\kappa_0^J(0)$ for $19\leq J\leq40$ ($\times$ points).
• Figure 6.4: $\kappa_0^J(1/4)$ for $J\leq18$ (solid points), their interpolating curve and its asymptotic value (solid curve and dashed line), and sampling estimates for $\kappa_0^J(1/4)$ for $19\leq J\leq40$ ($\times$ points).
• Figure 6.5: Difference between the normalized Mahler measure of the Turyn polynomial with relative shift $\alpha=1/4$, and its two companion Littlewood polynomials from Corollary \ref{['corTuryn']}, for primes $p<2000$, using \ref{['eqnMeasTLW']}. The black points show the result using $F_{p,\left\lfloor p/4\right\rceil}^+(x)$; the red ones illustrate those from $F_{p,\left\lfloor p/4\right\rceil}^-(x)$.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 3.1
• Corollary 3.2
• Lemma 4.1
• proof