Diffeomorphisms of the Circle and the Beurling--Helson Theorem

V. V. Lebedev

Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 1, pp. 30-35

Résumé

We consider the algebra $A(\mathbb{T})$ of absolutely convergent Fourier series on the circle $\mathbb{T}$. According to the Beurling–Helson theorem, the condition $\|e^{in\varphi}\|_A=O(1)$, $n\in\mathbb{Z}$, implies that $\varphi$ is trivial: $\varphi(t)=mt+\alpha$. We construct a nontrivial diffeomorphism $\varphi$ of $\mathbb{T}$ onto itself such that $\|e^{in\varphi}\|_A=O(\gamma(|n|)\log|n|)$, where $\gamma(n)$ is an arbitrary given sequence with $\gamma(n)\to+\infty$. By analogy with a conjecture due to Kahane, it is natural to suppose that this rate of growth is the slowest possible.