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Universal Radial Laws for Random Polynomials: Erdős Problem #522 and Non-Universal Fluctuations — arXiv:2610.11324

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.11324
License (as archived): CC0-1.0
Archived: 2026-10-09T06:28:21.393Z
SHA-256 of the archived text: 6fb034578e4c6596182e74f019f4d5bb53169d4950ebe4ef81cf2616121b01ec
Stable URL of this page: https://metraipolis.com/documento/dc_755dae3694ee

How to cite

Universal Radial Laws for Random Polynomials: Erdős Problem #522 and Non-Universal Fluctuations — arXiv:2610.11324 — arXiv (en), https://arxiv.org/abs/2610.11324 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_755dae3694ee (Metraipolis archive dc_755dae3694ee, sha256 6fb034578e4c6596182e74f019f4d5bb53169d4950ebe4ef81cf2616121b01ec)

BibTeX

@misc{dc755dae3694ee,
  title = {Universal Radial Laws for Random Polynomials: Erdős Problem #522 and Non-Universal Fluctuations — arXiv:2610.11324},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.11324},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_755dae3694ee; sha256 6fb034578e4c6596182e74f019f4d5bb53169d4950ebe4ef81cf2616121b01ec},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.11324. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We resolve Erdős Problem #522, posed in 1961, by proving that for random Littlewood polynomials the proportion of zeros in the closed unit disk converges almost surely to $1/2$. More generally, for partial sums of random power series with i.i.d. coefficients that are bounded, nondegenerate, and centrally symmetric, real Gaussian, or circular complex Gaussian, we prove that the proportion of zeros in the disk of radius $1+x/n$ converges almost surely to $\frac12(1+\coth x-1/x)$, uniformly in $x\in\mathbb{R}$. Thus almost all zeros lie at distance $O(1/n)$ from the unit circle. For random Littlewood polynomials, even with arbitrary dependence between degrees, the convergence at the unit circle holds at the almost-sure rate $O(n^{-1/4}\sqrt{\log n})$. We establish a law-of-the-iterated-logarithm criterion for dependent sequences and, for real Gaussian coefficients, obtain the sharp law of the iterated logarithm over all integer degrees, variance and cross-degree covariance asymptotics, quantitative central limit theorems, joint Gaussian limits, and increment bounds. For symmetric coefficient laws with bounded density and all moments finite, the number of zeros in the closed unit disk has variance $(c_G+\kappa_4/12)n+O_\xi(n^{399/400})$, and for random signs the constant is $c_G-1/6$. Thus the radial law is universal, while the fluctuations depend on the coefficient law through its fourth cumulant $\kappa_4$. All our main results are formally verified in Lean 4, and our proof of Erdős Problem #522 has been accepted as a solution to its formal statement in Google DeepMind's Formal Conjectures project.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.PR, nuovi lavori del giorno).
Categorie arXiv: Probability (math.PR) ; Classical Analysis and ODEs (math.CA); Complex Variables (math.CV).
Pagina del lavoro: https://arxiv.org/abs/2610.11324 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-09T05:53:01.613Z dalla pagina https://arxiv.org/list/math.PR/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).