On uniform convergence of Fourier series

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 $\phi :\mathbb{T}\to \mathbb{T}$ is a continuous piecewise linear but not linear map, then $\mathrm{\parallel }{e}^{in\phi }{\mathrm{\parallel }}_{U(\mathbb{T})}\simeq \log n$.