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Statement

Lemma 3.2. Fix an integer $n\ge1$ and complex inputs $Z_0,\ldots,Z_{n-1}$ in the closed unit disk. Each has a phase $d_j$ of modulus one with $Z_j=d_j|Z_j|$ (the phase of $0$ is set to $1$), and the half-sum of the defects is

$$\mu=\frac12\sum_{j=0}^{n-1}(1-|Z_j|).$$

There are $\eta_j\in\{d_j,-d_j\}$ such that

$$\sup_{t\in\mathbb T}\left|\sum_{j=0}^{n-1}(\eta_j-Z_j)\mathrm e(jt)\right| \le C\left(1+\sqrt{\mu\log(80n/\mu)}\right)$$

where $C$ is absolute. When $\mu=0$ the square-root term is taken to be zero, and in that case the error can be made zero. When every $Z_j$ is real the $\eta_j$ are signs, and shifting the frequency range from $0,\ldots,n-1$ to any $n$ consecutive integers changes nothing.

Source. OpenAI, Ultraflat real Littlewood polynomials, release folder preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026; TeX sections/rounding.tex lines 25--41 (label lem:complex-rounding), PDF pp. 4--5; proof p. 5 (sections/rounding.tex lines 43--115). Read 2026-10-07.

Read depth. Claims checked: the statement, and the statement of Lemma 3.1 it rests on, were read clause by clause in the TeX source. The proof was read for its structure (below) and no step was checked. Nothing here is independently reviewed.

Proof pointer

Section 3 (p. 5). With $p_j=(1-|Z_j|)/2\in[0,1/2]$, so $\sum p_j=\mu$, the manuscript forms a real matrix with $R=40n$ rows from the real and imaginary parts of $d_j\mathrm e(jt_\ell)$ on the grid $t_\ell=\ell/(20n)$ and rounds $p$ to a vector $p'\in\{0,1\}^n$ by a dyadic partial-coloring scheme: truncate $p$ to the grid $2^{-J}\mathbb Z$ with $n2^{-J}\le1$ (cost at most $1$), then at each stage $h=J,\ldots,1$ apply Lemma 3.1 (real matrix discrepancy, $\|A\xi\|_\infty\le C\sqrt{s\log(2R/s)}$ for $A\in[-1,1]^{R\times s}$, $s\le R$) to the $s_h$ coordinates whose current value is an odd multiple of $2^{-h}$, reversing all signs if needed so the total mass does not grow. Since each active coordinate carries mass at least $2^{-h}$, $s_h\le\min(n,2^h\mu)$, and monotonicity of $s\log(80n/s)$ makes the stage errors a geometric series summing to $C\sqrt{\mu\log(80n/\mu)}$, so that with the truncation cost $\|A(p'-p)\|_\infty\le1+C\sqrt{\mu\log(80n/\mu)}$. Setting $\eta_j=d_j(1-2p_j')$, the difference polynomial $Q$ obeys the bound at the grid points. To pass to the circle, with $S=\max_{|z|=1}|Q|$, the maximum principle for $Q$ and its reversal bounds $Q$ on the disk of radius $1+1/n$ by $\mathrm e\cdot S$, Cauchy's estimate bounds the derivative along the circle by $2\pi\mathrm e\,nS$, and since every point is within $1/(40n)$ of the grid and $\pi\mathrm e/20<1$, the grid bound absorbs the supremum. A unimodular monomial factor handles shifted frequency intervals, and real inputs give $d_j=\pm1$ and hence sign outputs.

Dependencies

Lemma 3.1 (real matrix discrepancy), quoted from the companion Nearly minimal maxima and positive minima of Littlewood polynomials, Lemma 6.2, which the companion derives from Spencer's partial-coloring method (Spencer 1985) and from Lovett and Meka's theorem for real vectors (Lovett--Meka 2015, Theorem 4 of arXiv:1203.5747v2); the maximum principle and Cauchy's estimate. External premises are taken at statement level; none was checked here.

Bears on

• Problem 1150: reaches the problem only through Theorem 1, where it rounds the real normalized coefficients of Proposition 5.1 to the claimed signs. Unverified here; the page's status rests on acceptance evidence.