---

Statement

Setting (pp. 3--4). $p_n(x)=\xi_0+\xi_1x+\cdots+\xi_nx^n$, with one coefficient sequence for all degrees (the distribution of $\xi_j$ does not depend on $n$), and $N_n(I)$ counts the real roots of $p_n$ in $I$.

Lemma 2.3 (p. 4). Let the coefficients $\xi_j$ be independent with zero mean, unit variance and uniformly bounded $(2+\epsilon)^{th}$ moments, and let $1\le n_1<n_2<\cdots$ be integers with $\inf_k n_{k+1}/n_k>1$. Then almost surely

$$\lim_{k\to\infty}\frac{N_{n_k}[0,1]}{\log(n_k)}=\frac{1}{2\pi}.$$

The lemma closes (quoted): "The same conclusion also holds for $[-1,0]$, $[1,\infty)$, $(-\infty,1]$ [sic], and $\mathbb{R}$."

Notes on that sentence. The interval $(-\infty,1]$ is as printed; the list of intervals in the proof (p. 6) has $(-\infty,-1]$ in its place. What the proof (p. 6) establishes for each interval $I$ of that list, which includes $\mathbb R$, is the almost sure limit $N_{n_k}(I)/\mathbb EN_{n_k}(I)\to1$; the constant in the limit of $N_{n_k}(I)/\log n_k$ is then the one in the expectation, and for $\mathbb R$ the paper recalls on p. 2 that $\mathbb EN_n=\frac2\pi\log n+O(1)$ under zero mean, unit variance and a finite $(2+\epsilon)^{th}$ moment (its reference [61]), not $\frac1{2\pi}\log n$.

Proof pointer

Section 2.1, pp. 5--6. Can and Nguyen's concentration estimate (the paper's reference [12], Theorems 1.4 and 1.5) bounds $\mathbf P(|N_n(I)-\mathbb EN_n(I)|\ge\epsilon\log n)$ by $\ll_\epsilon e^{-c_\epsilon\sqrt{\log n}}$, hence by $\ll_\epsilon(\log n)^{-2}$, for $I$ one of $\mathbb R$, $[0,1]$, $[1,\infty)$, $[-1,0]$, $(-\infty,-1]$. Along a lacunary sequence $\log n_k$ grows at least linearly in $k$, so these bounds are summable; Borel--Cantelli and $\epsilon\to0$ along a countable sequence give the limit.

Read depth

Claims checked: the statement and the proof in Section 2.1 were read clause by clause on the page images of the print. The Can--Nguyen estimate is cited, not proved, in the paper and was not read. Nothing here is independently reviewed.

Dependencies

The concentration estimate of Can and Nguyen, external to the corpus. Used by Theorem 1.1.

Source. Yen Q. Do, A strong law of large numbers for real roots of random polynomials, arXiv:2403.06353 (2024); the edition read is named on the source card.

Bears on

• Problem 521: the lemma's closing sentence includes $\mathbb R$, and its proof, written out for $[0,1]$ and said to be the same for the other intervals, gives, for independent uniform signs $\pm1$, the almost sure limit $N_{n_k}(\mathbb R)/\mathbb EN_{n_k}(\mathbb R)\to1$ along every sequence of degrees with $\inf_k n_{k+1}/n_k>1$. This concerns lacunary degree sequences only; the paper does not extend it to all $n$.