Existence of degree-n ±1 polynomials P with δ√n ≤ |P(z)| ≤ Δ√n for all |z|=1 and absolute constants Δ > δ > 0.
Ruzsa, Negative values of cosine sums, Acta Arith., 111 (2004), 179–186
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Flat Littlewood Polynomials Exist
Existence of degree-n ±1 polynomials P with δ√n ≤ |P(z)| ≤ Δ√n for all |z|=1 and absolute constants Δ > δ > 0.