Turyn's conjecture that binary merit factors are bounded

Section 8 deduces from Theorem 1.1 (real-sign polynomials of every large length with maximum $(1+o(1))\sqrt N$) that $\max_{A\in\{-1,1\}^N}F(A)\to\infty$ as $N\to\infty$ through all integers, disproving Turyn's conjecture in this all-length form. Via Downarowicz and Lacroix it also gives a uniquely ergodic binary Morse shift with simple spectrum and absolutely continuous zero-coordinate spectral measure with $L^2$ density. The construction is existential: no explicit sequences, no rate and no algorithm are given.

Section 8 deduces from Theorem 1.1 (real-sign polynomials of every large length with maximum $(1+o(1))\sqrt N$) that $\max_{A\in\{-1,1\}^N}F(A)\to\infty$ as $N\to\infty$ through all integers, disproving Turyn's conjecture in this all-length form. Via Downarowicz and Lacroix it also gives a uniquely ergodic binary Morse shift with simple spectrum and absolutely continuous zero-coordinate spectral measure with $L^2$ density. The construction is existential: no explicit sequences, no rate and no algorithm are given.