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Statement

Setting (pp. 1--2). $P(z)=\sum_{k=0}^{n-1}X_kz^k$ with independent uniform signs $X_k$, as in Theorem 1. Write $d\mu(x)=dx/(2\pi)$ on $[-\pi,\pi]$, $\sigma(r)^2=\mathbb E|P(re^{i\theta})|^2=\sum_{k=0}^{n-1}r^{2k}$ (so $\sigma(1)^2=n$), and $\widetilde P(re^{i\theta})=P(re^{i\theta})/\sigma(r)$.

Lemma 1.3 (p. 2). Let $\ell\in\{1,2\}$. For every $r\in[1-n^{-11/10},1+n^{-11/10}]$, as $n\to\infty$,

$$\mathbb E\Bigl[\Bigl(\int_{-\pi}^{\pi}\log|\widetilde P(re^{i\theta})|\,d\mu(\theta)\Bigr)^{\ell}\Bigr] =\Bigl(-\frac\gamma2\Bigr)^{\ell}+O\Bigl(\frac{(\log n)^2}{\sqrt n}\Bigr),$$

where $\gamma$ is Euler's constant. The constants are uniform in $r$ over this range (p. 5).

At $r=1$ the integral is $\log(M(P)/\sqrt n)$, $M$ the Mahler measure. The paper notes that the case $r=1$, $\ell=1$ extends to $P$ itself the limit $\mathbb E\log(M(\widehat P)/\sqrt n)\to-\gamma/2$ that Choi and Erdélyi proved for the truncation $\widehat P=\max\{|P|,n^{-1}\}$, and that the case $r=1$ with Chebyshev's inequality gives $M(P)/\sqrt n\to e^{-\gamma/2}$ in probability (pp. 2--3).

Source. Oren Yakir, Approximately half of the roots of a random Littlewood polynomial are inside the disk, arXiv:2011.06234v2 (2022); published in Studia Math. 261 (2021), 227--240. Labels and pages here are those of arXiv v2: Lemma 1.3 on p. 2, its proof in Section 3 on pp. 5--8, with Proposition 3.1 proved in Section 4 (pp. 8--11) and Lemma 3.2 in Section 5 (pp. 11--13). The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 3, pp. 6--8; the paper writes out only $\ell=2$. With the event $A_\theta=\{|P(re^{i\theta})|\le n^{-A}\}$ for a large constant $A$, the second moment is split by Fubini into the parts where neither, one, or both of $A_\theta$, $A_\varphi$ occur. The parts meeting $A_\theta$ or $A_\varphi$ are $O(n^{-1/2})$, by Cauchy--Schwarz, a deterministic bound of order $n^4$ on the fourth moment of $\log|\widetilde P|$ over the circle, and the small-ball estimate of Proposition 3.1. On the main part, a set of measure $O(n^{-1/2})$ near the diagonal and the axes is discarded, and off it the pair $(|\widetilde P(re^{i\theta})|^2,|\widetilde P(re^{i\varphi})|^2)$ is compared with two independent exponential variables through Lemma 3.2, a Berry--Esseen bound; for the exponential law, $\int_0^\infty(\log x)e^{-x}\,dx=-\gamma$.

Dependencies

Proposition 3.1 (p. 6): for $a\in(0,1/3)$ and $r$ in the range above, $\int_{-\pi}^{\pi}\mathbb P(|P(re^{i\theta})|\le a),d\mu(\theta)\le C(n^{-5}+n^{240}a\log(1/a))$; the paper says it is essentially borrowed from Ibragimov and Zeitouni and proves it in Section 4 from Esseen's concentration inequality and Turán's lemma (Lemma 4.1, p. 8). Lemma 3.2 (p. 6): the distribution function of $|\widetilde P(re^{i\theta})|^2$ is within $C/\sqrt n$ of $1-e^{-x}$ for $|\theta|\ge n^{-1/2}$, and the joint distribution function at two angles outside $[-n^{-1/2},n^{-1/2}]$ at distance more than $n^{-1/2}$ is within $C/\sqrt n$ of the product.

Bears on

• Problem 522: Lemma 1.3 is the concentration input from which the paper derives Theorem 1, the in-probability form of the root count the problem asks about; the lemma itself says nothing about roots.