A proof of Kahane's conjecture in the Wiener algebra

Abstract

We prove Kahane's conjecture: if a continuous unimodular function $F$ on the circle satisfies $\|F^n\|_{A(\mathbb{T})}=o(\log|n|)$ as $|n|\to\infty$, $n\in\mathbb{Z}$, then $F(x)=e^{i(mx+b)}$ for some $m\in\mathbb{Z}$ and $b\in\mathbb{R}$. The main step is a logarithmic lower bound for averages: for every nonconstant real-valued $φ\in A(\mathbb{T})$, $\frac{1}{T}\int_a^{a+T}\|e^{itφ}\|_{A(\mathbb{T})}dt \ge c_φ\log T \quad(a\in\mathbb{R},\ T\ge T_φ)$, where $c_φ>0$ and $T_φ>1$ are independent of $a$. The proof uses exact sampling on finite subgroups of the circle, uniformly bounded localized modulations, and the weighted non-harmonic Littlewood inequality of Jaming--Kellay--Saba. A suitable cutoff and a small linear perturbation of the phase ensure that a fixed positive proportion of the coefficients at the beginning of the frequency ordering have modulus one. Passing to integer powers and removing the degree of the circle map yield Kahane's conjecture without any additional smoothness assumptions. We also give equivalent phase, convolution, and operator formulations, and derive consequences for homomorphisms of weighted Wiener algebras and for higher-dimensional tori.