The quantitative Beurling-Helson Theorem

TL;DR

This paper establishes a quantitative condition under which a continuous function on the circle must be affine with integer slope, based on the growth rate of a certain Fourier norm.

Contribution

It provides a new quantitative criterion linking the Fourier norm growth to the linearity of functions on the circle, refining the classical Beurling-Helson theorem.

Findings

If the Fourier norm grows slower than a logarithmic rate, the function must be affine with integer slope.

The result quantifies the classical theorem by specifying the growth rate threshold.

The theorem applies to continuous functions on the circle with controlled Fourier behavior.

Abstract

We show that for any $\varepsilon>0$ if $\phi:\mathbb{T} \rightarrow \mathbb{T}$ is continuous and $\|\exp(-2\pi i z \phi)\|_{A(\mathbb{T})} =O_{|z|\rightarrow \infty}(\log^{\frac{1}{8}-\varepsilon} |z|)$ then $\phi(x)=wx+t$ for some $w \in\mathbb{Z}$ and $t \in \mathbb{T}$.