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Math-5711-A1 · revision r1 · on Math-5711

Reused writeup on Math-5711 (gpt pro 5.2)

No claim recorded · LLM-generated · gpt pro 5.2 ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Erdős Problem 522 --- collection record
  2. PROVENANCE AND STATUS
  3. REUSED SOURCE 1
  4. COMPLETION ESTIMATE

Erdős Problem 522 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/522

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/522.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/522.tex.

FORMAL RESTATEMENT. Let $\epsilon_0,\dots,\epsilon_n$ be i.i.d. uniform in $\{-1,1\}$ and \[f_n(z)=\sum_{k=0}^n \epsilon_k z^k. \] Let $R_n$ be the number of complex roots of $f_n$ in the closed unit disk $\{z\in\mathbb C:|z|\le 1\}$, counted with multiplicity. Question: Is it true that, with probability $1$, \[\frac{R_n}{n/2}\to 1\qquad (n\to\infty)? \]

QUICK LITERATURE/CONTEXT CHECK. The problem statement reports that Yakir proved the weaker claim $R_n/(n/2)\to 1$ in probability, and even that \[\lim_{n\to\infty}\mathbb P(|R_n-n/2|\ge n^{9/10})=0. \]

ATTACK PLAN.

  • Proof track idea 1: Upgrade Yakir's convergence-in-probability to almost sure convergence by proving summable tail bounds for deviations $|R_n-n/2|$ on a suitable subsequence and controlling interpolation.
  • Proof track idea 2: Use an argument principle representation for $R_n$ in terms of $\arg f_n(e^{i\theta})$ and prove a strong law for this winding number.
  • Disproof track: Search for an event (occurring infinitely often) that forces $\Omega(n)$ roots to concentrate inside or outside the unit disk (e.g. polynomials with atypically large $|f_n|$ on $|z|=1$ that bias the zero distribution).

WORK.

Fast reality check (small $n$ enumeration; Monte Carlo). Using the same computations as in Problem 521, we counted the number of roots in the closed unit disk (numerically, tolerance $|r|\le 1+10^{-8}$). 

Exact enumeration:
 n=4: avg_in=2.375, min_in=1, max_in=4
 n=6: avg_in=3.25,  min_in=1, max_in=6
 n=8: avg_in=4.3984375, min_in=1, max_in=8
 n=10: avg_in=5.09375, min_in=1, max_in=10
Monte Carlo (200 samples):
 n=20: avg_in=10.17
 n=50: avg_in=25.145
 n=100: avg_in=49.77
Ratios $\texttt{avg\_in}/(n/2)$ in these experiments were close to $1$.

Lemma 522.1 (reversal symmetry and an expectation identity). For any $n\ge 0$, define the reversed polynomial \[f_n^*(z)=z^n f_n(1/z)=\sum_{k=0}^n \epsilon_k z^{n-k}. \] Let $R_n^{<}$ (resp. $R_n^{>}$, $R_n^{=}$) denote the number of roots of $f_n$ with $|z|<1$ (resp. $|z|>1$, $|z|=1$), counted with multiplicity. Then \[\mathbb E R_n^{<} = \mathbb E R_n^{>} = \frac{n-\mathbb E R_n^{=}}{2}. \]

Proof. First, $f_n^*$ has coefficient vector $(\epsilon_n,\dots,\epsilon_0)$ and thus the same distribution as $f_n$ (i.i.d. coefficients).

Next, $f_n(0)=\epsilon_0\ne 0$, so $0$ is not a root and all roots are nonzero. If $z$ is a root of $f_n$ of multiplicity $m$, then $1/z$ is a root of $f_n^*$ of multiplicity $m$ (this is immediate from $f_n^*(w)=w^n f_n(1/w)$). Therefore the multiset of moduli is inverted: $|z|<1$ for $f_n$ corresponds to $|1/z|>1$ for $f_n^*$. Hence, for every realization, \[R_n^{<}(f_n)=R_n^{>}(f_n^*). \] Taking expectations and using $f_n\stackrel{d}{=} f_n^*$ gives \[\mathbb E R_n^{<} = \mathbb E R_n^{>}. \] Finally, since a degree-$n$ polynomial has exactly $n$ roots counting multiplicity, we have the deterministic identity \[R_n^{<}+R_n^{=}+R_n^{>}=n. \] Taking expectations and substituting $\mathbb E R_n^{<}=\mathbb E R_n^{>}$ yields the claimed formula. $\square$

Lemma 522.2 (deterministic root annulus). For every realization of $(\epsilon_k)_{0\le k\le n}$, every root $z$ of $f_n$ satisfies $\tfrac12\le |z|\le 2$.

Proof. This is exactly Lemma 521.1 applied to $f_n$. $\square$

VERIFICATION.

  • Lemma 522.1: The key step is that $f_n^*$ has the same distribution as $f_n$; this uses only i.i.d. of the coefficients. Also, $f_n(0)\ne 0$ ensures inversion $z\mapsto 1/z$ is valid for every root.
  • The identity $R_n^{<}+R_n^{=}+R_n^{>}=n$ uses only the fundamental theorem of algebra, counting multiplicity.

FINAL. UNRESOLVED

(i) Strongest proved partial result: An exact expectation identity $\mathbb E R_n^{<}=\mathbb E R_n^{>}=(n-\mathbb E R_n^{=})/2$ (Lemma 522.1), and deterministic root localization $\tfrac12\le |z|\le 2$ (Lemma 522.2).

(ii) First gap (crisp): Control $R_n^{=}$ (roots exactly on the unit circle) and, more importantly, upgrade convergence in probability of $R_n$ to almost sure convergence, i.e. prove that \[\sum_{n=1}^\infty \mathbb P\big(|R_n-n/2|\ge \eta n\big)<\infty\quad\text{for every fixed }\eta>0 \] or otherwise obtain an almost sure argument.

(iii) Top 3 next moves (concrete):

  1. Prove a quantitative tail bound $\mathbb P(|R_n-n/2|\ge n^{\alpha})\le n^{-2}$ for some $\alpha<1$ (even on a sparse subsequence) to apply Borel--Cantelli.
  2. Bound the expected number (or probability) of unit-circle roots $R_n^{=}$ sharply (e.g. show $\mathbb E R_n^{=} = o(n)$), which together with Lemma 522.1 would imply $\mathbb E R_n\sim n/2$.
  3. Computation: for moderate $n$ (say $n\le 40$) empirically estimate $\mathbb P(|R_n-n/2|\ge n^{0.9})$ via Monte Carlo to guess the correct deviation scale.

(iv) Minimal counterexample structure: A counterexample to almost sure convergence would likely manifest as infinitely many $n$ for which $|R_n-n/2|\ge c n$ (linear deviations). This would require persistent, recurring structure in the random coefficients that biases the zero set substantially inside/outside the unit circle.

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.