  We consider the space $U(\mathbb T)$ of all continuous functions on the
circle $\mathbb T$ with uniformly convergent Fourier series. We show that if
$\varphi: \mathbb T\rightarrow\mathbb T$ is a continuous piecewise linear but
not linear map, then $\|e^{in\varphi}\|_{U(\mathbb T)}\simeq\log n$.
