Erdős’s ultraflat Littlewood polynomials conjecture

Show that ultraflat Littlewood polynomials do not exist; equivalently, prove the existence of an absolute constant c > 0 such that for all n, the flatness measure C_golay^w(n) ≥ c for Littlewood polynomials of degree n.

Background

Littlewood polynomials with coefficients ±1 are studied for flatness on the unit circle. While flat polynomials exist, “ultraflat” (asymptotically perfectly flat) are conjectured not to.

The conjecture’s resolution would impact signal design, combinatorial constructions, and analytic bounds on trigonometric polynomials.

References

In 1962 Erd\H{o}s conjectured that ultraflat Littlewood polynomials do not exist, so that $C_{\ref{golay}w(n) \geq c$ for some absolute constant $c>0$.

— Mathematical exploration and discovery at scale  (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Flat polynomials and Golay’s merit factor conjecture” (Section 4.16)