Erdos's question on the maximum modulus of real Littlewood polynomials: is it at least (1 + c) sqrt(N)?

Theorem 1.1: $\min_{\epsilon}\max_{|z|=1}|\sum_{k<N}\epsilon_k z^k|=(1+o(1))\sqrt N$ as $N\to\infty$ through all integers, so no constant $c>0$ as in Erdos's question exists. The construction relaxes to real coefficients in $[-1,1]$ with small defect, using sampled quadratic-phase modes and a hypergraph interval packing, then rounds to signs by a Lovett-Meka partial-coloring argument. It is existential and gives no rate. The principal paper gives no uniform lower bound; the October 5 companions add $|P|\ge\sqrt N/16$ and then full two-sided ultraflatness $(1\pm\epsilon)\sqrt N$ at every large length.

This result conflicts with published claims. Preprints by el Abdalaoui (arXiv:1609.03435 and two others) claim the opposite, that such flat polynomials do not exist; the release's manuscript argues in its Appendix A that those arguments have specific gaps. Until the conflict is settled in public, the entry is Contested. Theorem 1.1: $\min_{\epsilon}\max_{|z|=1}|\sum_{k<N}\epsilon_k z^k|=(1+o(1))\sqrt N$ as $N\to\infty$ through all integers, so no constant $c>0$ as in Erdos's question exists. The construction relaxes to real coefficients in $[-1,1]$ with small defect, using sampled quadratic-phase modes and a hypergraph interval packing, then rounds to signs by a Lovett-Meka partial-coloring argument. It is existential and gives no rate. The principal paper gives no uniform lower bound; the October 5 companions add $|P|\ge\sqrt N/16$ and then full two-sided ultraflatness $(1\pm\epsilon)\sqrt N$ at every large length.