Let (epsilon_k)_k≥ 0 be independently uniformly chosen at random from -1,1. If R_n counts the number of real roots of f_n(z)=∑_0≤ k≤ nepsilon_k z^k then is it true that, almost surely, lim_n→ ∞fracR_nlog n=frac2pi?

Erdős Problem #521 — Let (epsilon_k)_k≥ 0 be independently uniformly chosen at random from -1,1.

The problem asks whether the real-root count has almost-sure asymptotic constant $2/\pi$ . Erdős and Offord established the corresponding leading term without almost-sure convergence in 1956.

Known results

Erdős–Offord, 1956: the expected-scale asymptotic is $(2/\pi+o(1))\log n$ .
Do, 2024: almost surely, the roots in $[-1,1]$ satisfy $R_n[-1,1]/\log n\to1/\pi$ .
Earlier work established $\mathbb{E}R_n=(2/\pi)\log n+O(1)$ , but not the required almost-sure law.

2024 strong law

A 2024 arXiv paper proves almost-sure convergence for real roots of Kac polynomials with iid coefficients having zero mean, unit variance, and bounded $(2+\epsilon)$ -th moment. Its full-real-line conclusion applies to iid random signs and gives $R_n/\log n\to2/\pi$ almost surely, settling Problem #521.

Current status (as of March 2026): the almost-sure assertion for random Littlewood polynomials is settled by the 2024 strong-law theorem.

Erdős Problem #521 — Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$ .

Primary source

Additional references

Progress summary

Solutions 0