Research Problems in Function Theory — Problem 4.16

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$.Use the family $P_\epsilon(z)=\sum_{k=1}^n\epsilon_kz^k$, $\epsilon\in\{-1,1\}^n$, consisting of $2^n$ polynomials. Put \[m(P_\epsilon)=\min_{|z|=1}|P_\epsilon(z)|. \]

Statement recorded in the supplied source.

Does \[2^{-n}\#\{\epsilon\in\{-1,1\}^n:m(P_\epsilon)\ge1\}\longrightarrow0\quad(n\to\infty)?\] If this is false, determine the correct typical asymptotic behaviour of the minimum modulus, including its scale or limiting bounds for all but $o(2^n)$ sign choices. [S2]

Short English statement

Does almost every plus-or-minus-one polynomial have modulus below one somewhere on the unit circle? If not, determine the correct typical minimum behaviour.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304016 (AMR-022-4016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 4.16, its complete Update 4.16, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.