Erdős problem 228 · problems.science

Does there exist, for all large $n$ , a polynomial $P$ of degree $n$ , with coefficients $\pm1$ , such that $\sqrt n \ll |P(z)| \ll \sqrt n$ for all $|z|=1$ , with the implied constants independent of $z$ and $n$ ?

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228.lean

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The answer is yes, proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].

[BBMST19] Balister, P. and BollobÁs, B. and Morris, R. and Sahasrabudhe, J. and Tiba, M., _Flat Littlewood Polynomials Exist_. arXiv:1907.09464 (2019).

FormalConjectures/ErdosProblems/228.leanErdos228.erdos_2287 linesExact file

1True ↔2  ∃ c₁ c₂,3    ∀ᶠ (n : ℕ) in Filter.atTop,4      ∃ p,5        p.degree = ↑n ∧6          (∀ i ≤ n, p.coeff i = 1 ∨ p.coeff i = -1) ∧7            ∀ (z : ℂ), ‖z‖ = 1 → √↑n < c₁ * ‖Polynomial.eval z p‖ ∧ ‖Polynomial.eval z p‖ < c₂ * √↑n

SolvedStatement only, no proof

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