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Multifractal analysis of the divergence of Fourier series

A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$, $p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every $x\in \mathbb T$. Beside this property, the series may diverge at some point, without exceeding $O(n^{1/p})$. We define the divergence index at $x$ as the infimum of the positive real numbers $β$ such that $S_nf(x)=O(n^β)$ and we are interested in the size of the exceptional sets $E_β$, namely the sets of $x\in\mathbb T$ with divergence index equal to $β$. We show that quasi-all functions in $L^p(\TT)$ have a multifractal behavior with respect to this definition. Precisely, for quasi-all functions in $L^p(\mathbb T)$, for all $β\in[0,1/p]$, $E_β$ has Hausdorff dimension equal to $1-βp$. 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.