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Statement

Setting (p. 1). $\mathbb U_n$ is the set of polynomials $F(z)=\sum_{k=0}^n a_kz^k$ with every $a_k=\pm1$.

Conjecture (p. 9). The paper says its computations strongly suggest that there are constants $0<\delta<C$, even with $\delta=0.5$ and $C=1.5$, such that for all large $n$ there exists $F\in\mathbb U_n$ with

$$\delta<|F(z)|/\sqrt{n+1}<C$$

for $z$ on the unit circle (inequality (16)).

In the same section (p. 9) the paper recalls that Beck proved, by a non-constructive argument, that polynomials satisfying (16) exist for some positive $\delta$, $C$ when the coefficients are required to satisfy $a_k^{400}=1$, and conjectures that for each integer $r\ge2$, with $r$-th roots of unity as coefficients, the limits corresponding to $M$ and $m$ exist and tend to $1$ as $r\to\infty$.

Scope

Stated as what the data suggest, not proved. The degree-12 Barker polynomial (Fig. 3, p. 7) and the degree-94 skew-symmetric polynomial of Fig. 5 (p. 11) are drawn with circles of radii 0.5 and 1.5; finite examples do not establish the all-large-$n$ statement.

Read depth

Claims checked: the statement was read on the page image of the print. Nothing here is independently reviewed.

Dependencies

None.

Source. Andrew Odlyzko, "Search for Ultraflat Polynomials with Plus and Minus One Coefficients," in Connections in Discrete Mathematics, pp. 39--55, Cambridge University Press, 2018, doi:10.1017/9781316650295.004; the version read, the author's revised version of 18 May 2017, and its page numbering are named on the source card.

Bears on

• Problem 228: (16) for all large $n$ is the affirmative answer to the problem's question, in the paper's normalization by $\sqrt{n+1}$ rather than $\sqrt n$; the paper conjectures it and proves nothing toward it.