Multifractal analysis of the divergence of Fourier series

A famous theorem of Carleson says that, given any function $f\in L^p(\text{TT})$, $p\in (1,+\mathrm{\infty })$, its Fourier series $(S_{n}f(x))$ converges for almost every $x\in \mathbb{T}$. Beside this property, the series may diverge at some point, without exceeding $O(n^{1\mathrm{/}p})$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f(x)=O(n^{\beta })$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in \mathbb{T}$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p(\text{TT})$ have a multifractal behavior with respect to this definition. Precisely, for quasi-all functions in $L^p(\mathbb{T})$, for all $\beta \in [0,1\mathrm{/}p]$, $E_{\beta }$ has Hausdorff dimension equal to $1-\beta p$. We also investigate the same problem in $\mathcal{C}(\mathbb{T})$, replacing polynomial divergence by logarithmic divergence. In this context, the results that we get on the size of the exceptional sets are rather surprizing.