I know that there exist continuous function $f: [0,2\pi]\rightarrow\mathbb{R}$ whose Fourier series diverges at all rational points of $[0,2\pi]$(c.f. Katznelson).We also know that the set of divergence of $f$ must be of measure zero, from Carleson, since a continuous function is undoubtedly in $L^2([0,2\pi])$. But there is a gap: can a continuous function have its Fourier series diverges on an uncountable set of measure zero(like the Cantor set)?
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Kahane and Katznelson proved that given any set $E$ of measure zero, there exists a continuous function whose Fourier series diverges on $E$ (see this paper). 

