Minimum of the absolute value of random trigonometric polynomials with coefficients \(\pm 1\) (Q1905284)

If \(\xi_0, \dots, \xi_{n - 1}\) are \(n\) independent random variables that are equal to \(+ 1\) or \(- 1\) with equal probability \(1/2\) and if \(\text{Prob} (\min_{x \in T} |\sum^{n - 1}_{j = 0} \xi_j \exp (ijx) |> u)\) is denoted by \(P_n (u) = P(u)\), then Littlewood has conjectured that \(P (\varepsilon \sqrt n) \to 0\) as \(n \to \infty\) and Kashin has established that \(P (\sqrt n (\log n)^{- 1/3}) \to 0\) as \(n \to \infty\). The author proves Odlyzko's conjecture that for large \(n\) and any \(\varepsilon > 0\) most of the polynomials \(T(x) = \sum^{n - 1}_{j = 0} \pm \exp (ijx)\) satisfy the inequality \(\min |T(x) |< n^{- 1/2 + \varepsilon}\).