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  "body": "\n## Statement\n\nLet $E \\subset \\R$ be a set of Lebesgue measure zero, that is, a [[null set|VQQM4Y550T]].\nThen there is a [[continuous|0X3WGX95FE]] $2\\pi$-periodic function $f \\colon \\R \\to \\C$ whose [[Fourier series|G12KXDHPPS]] diverges at every point of $E$.\n\n## Remarks\n\nKahane and Katznelson proved this in 1966.\nTogether with [[Carleson's theorem|FBH6KG484Z]] it describes the sets of divergence exactly.\nA set lies in the divergence set of the Fourier series of some continuous function if and only if it has measure zero.\nThe proof builds $f$ as a uniformly convergent sum of trigonometric polynomials, using covers of $E$ by intervals of small total length.\nFor a single point $E = \\set{0}$ the result is the classical [[example of du Bois-Reymond|FPDWWD12FE]].\nFor example, $E$ can be the set of rational multiples of $\\pi$, or the Cantor set.\nThen some continuous function has a Fourier series diverging at each point of $E$.\n\n## Explanations\n\n### L1 human\n\nCarleson's theorem says that the Fourier series of a continuous function can fail to converge only on a set of total length zero.\nThis theorem says that every set of length zero really occurs: one can choose a continuous function whose Fourier series fails to converge at each point of that set.\nSo nothing better than \"almost everywhere\" can be true.\n\n### L2 human\n\nThe result shows that Carleson's theorem is sharp for continuous functions.\nIt contrasts with smooth or Hölder continuous functions, whose Fourier series converge everywhere by the [[Dini test|F5PHA35M50]].\nSo the divergence sets of continuous functions are as large as Carleson's theorem allows.\n",
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