Erdős Problem #230

Let \(P(z)=\sum_{1\leq k\leq n}a_kz^k\) for some \(a_k\in \mathbb{C}\) with \(\lvert a_k\rvert=1\) for \(1\leq k\leq n\). Does there exist a constant \(c>0\) such that, for \(n\geq 2\), we have \[\max_{\lvert z\rvert=1}\lvert P(z)\rvert \geq (1+c)\sqrt{n}?\] This is Problem 4.31 in [Ha74], in which it is described as a conjecture of Erdős and Newman. The lower bound of \(\sqrt{n}\) is trivial from Parseval's theorem. Körner [Ko80] constructed, for all \(n\geq 2\), polynomials \(P(z)=\sum_{k\leq n} a_kz^k\) with \(\lvert a_k\rvert=1\) for \(1\leq k\leq n\) such that, for all \(z\) with \(\lvert z\rvert=1\), \[(c_1-o(1))\sqrt{n} \leq \lvert P(z)\rvert \leq (c_2+o(1))\sqrt{n}\] for some absolute constants \(0<c_1\leq c_2\). The answer is no (contrary to Erdős' initial guess). Kahane [Ka80] constructed 'ultraflat' polynomials \(P(z)=\sum a_kz^k\) with \(\lvert a_k\rvert=1\) such that \[P(z)=(1+o(1))\sqrt{n}\] uniformly for all \(z\in\mathbb{C}\) with \(\lvert z\rvert=1\), where the \(o(1)\) term \(\to 0\) as \(n\to \infty\). For more details see the paper [BoBo09] of Bombieri and Bourgain and where Kahane's construction is improved to yield such a polynomial with \[P(z)=\sqrt{n}+O(n^{\frac{7}{18}}(\log n)^{O(1)})\] for all $z\