Research Problems in Function Theory — Problem 4.15

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 $\mathbb T=\partial\mathbb D$.For each $n\ge1$, consider all $2^n$ polynomials \[P_\epsilon(z)=\sum_{k=1}^n\epsilon_kz^k,\qquad\epsilon=(\epsilon_1,\ldots,\epsilon_n)\in\{-1,1\}^n. \] Let $Z_{\mathbb D}(P_\epsilon)$ be the number of zeros in the open disc $\mathbb D$, counted with multiplicity and including the zero at the origin.

Statement recorded in the supplied source.

Is it true that for every $\delta>0$, \[2^{-n}\#\left\{\epsilon\in\{-1,1\}^n:\left|Z_{\mathbb D}(P_\epsilon)-\tfrac n2\right|>\delta n\right\}\longrightarrow0\quad(n\to\infty)?\] Equivalently, all but $o(2^n)$ of these polynomials have $n/2+o(n)$ zeros in $\mathbb D$, with the error interpreted uniformly outside a vanishing fraction of sign choices. [S2]

Short English statement

For almost every sign choice, does a polynomial with coefficients plus or minus one and no constant term have about half its zeros inside the unit disc?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304015 (AMR-022-4015). 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 4.15, its complete Update 4.15, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.