A disproof of $L^\alpha$ polynomials Rudin conjecture, $2 \leq \alpha<4.$

TL;DR

This paper disproves Rudin's conjecture on $L^eta$-norms of polynomials for certain $eta$, providing counterexamples inspired by Bourgain's work on NLS and establishing measure-zero properties of specific sets.

Contribution

It presents a counterexample to Rudin's $L^eta$ polynomial conjecture for $2 \,\leq\, \alpha < 4$, and offers an alternative proof of Cordoba's theorem using Paley-Littlewood inequalities.

Findings

Rudin's conjecture on $L^eta$-norms fails for $2 \leq \alpha < 4$.

Constructs a set of measure zero where the Weyl sum exceeds bounds infinitely often.

Provides an alternative proof of Cordoba's theorem based on harmonic analysis inequalities.

Abstract

It is shown that the $L^\alpha$-norms polynomials Rudin conjecture fails. Our counterexample is inspired by Bourgain's work on NLS. Precisely, his study of the Strichartz's inequality of the $L^6$-norm of the periodic solutions given by the two dimension Weyl sums. We gives also a lower bound of the $L^\alpha$-norm of such solutions for $\alpha \neq 2$. As a consequence, we establish that for any $0<a<b,$ the following set $E(a,b)=\Big\{(x,t) \in \mathbb{T}^2 \; : \;a \sqrt{N} \leq \Big|\sum_{n=1}^{N}e(n^2t+nx)\Big| \leq b \sqrt{N} \; \textrm{~infinitely~often}\;\Big\},$ has a Lebesgue measure $0$. We further present an alternative proof of Cordoba's theorem based on Paley-Littlewood inequalities.