Claims

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1980_09_01_kahane: Kahane's 1980 theorem: for every n there is a polynomial of degree n with complex coefficients of modulus one whose modulus on the unit circle is (1+o(1)) times the square root of n uniformly, so the answer is no.

2009_06_30_bombieri_bourgain: Theorems 4 and 7 of Bombieri and Bourgain: unimodular polynomials whose modulus on the unit circle is the square root of n up to an error of order n to the power 7/18 plus epsilon, also by an effective construction.

2026_09_23_openai: Theorem 1.1 of the OpenAI release manuscript of 23 September 2026: for every positive eta and every large N some signs plus or minus one give a polynomial of maximum modulus at most (1+eta) times the square root of N.

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