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Statement

Setting (pp. 1--3). $\mathbb U_n$ is the set of polynomials $F(z)=\sum_{k=0}^n a_kz^k$ with every $a_k=\pm1$. For such $F$, $M(F)=\max_{|z|=1}|F(z)|/\sqrt{n+1}$, $m(F)=\min_{|z|=1}|F(z)|/\sqrt{n+1}$ and $W(F)=M(F)-m(F)$ (equation (3), p. 2), and $M_n=\min_{F\in\mathbb U_n}M(F)$, $m_n=\max_{F\in\mathbb U_n}m(F)$, $W_n=\min_{F\in\mathbb U_n}W(F)$ (equations (5)--(7), pp. 2--3). By Parseval, $\|F\|_2^2=n+1$ (equation (2), p. 2), so $M(F)\ge1\ge m(F)$.

Conjecture (p. 4, equations (8)--(10)). Each of the limits $\lim_{n\to\infty}M_n=M$, $\lim_{n\to\infty}m_n=m$ and $\lim_{n\to\infty}W_n=W$ exists.

The paper states the conjecture on pp. 3--4 as the outcome of its searches, and adds (p. 4) that the computations suggest $M\approx1.27$, $m\approx0.64$ and $W\approx0.79$; p. 5 says these values were derived from the computed skew-symmetric values $M_n^*$, $m_n^*$, $W_n^*$ (see the conjecture on p. 5).

Consequences the paper draws (p. 4). These conjectures would imply that ultraflat polynomials in $\mathbb U_n$ do not exist, and that Golay--Rudin--Shapiro polynomials are far from optimal in terms of never being large. The existence of $W$ together with $W<1$ would give constants $0<c_1<c_2$ such that for all high degrees there are $F\in\mathbb U_n$ with $c_1<m(F)<M(F)<c_2$, which the paper identifies as Littlewood's conjecture $(C_1)$.

Scope

A conjecture supported by computation, not a theorem. The evidence is the exhaustive search through degree 52 and the skew-symmetric search through degree 104 recorded on the search page. The rigorous facts the paper recalls beside it are $M_n\le\sqrt2$ for $n=2^k-1$ from Golay--Rudin--Shapiro polynomials and boundedness of $M_n$ over all $n$ (p. 3); the paper says (p. 3) it is not even known whether $\limsup_{n\to\infty}m_n>0$.

Read depth

Claims checked: the definitions and the conjecture with its estimates were read on the page images 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 1150: the problem asks for a fixed $c>0$ with $\max_{|z|=1}|P(z)|>(1+c)\sqrt n$ for every $\pm1$ polynomial $P$ of every large degree $n$. The conjecture that $M_n\to M$ with $M\approx1.27$, if true with $M>1$, would answer it yes, any $c<M-1$ serving for large $n$ after the factor $\sqrt{(n+1)/n}$. The paper proves no lower bound on $M_n$ beyond $M_n\ge1$.
• Problem 228: the problem asks for $\pm1$ polynomials of every large degree $n$ with $\sqrt n\ll|P(z)|\ll\sqrt n$ on the unit circle. The paper notes that the existence of $W$ with $W<1$ would give such polynomials (Littlewood's $(C_1)$); it proves neither.