A continuous Helson surface in 𝐑 3
Annales de l'Institut Fourier, Volume 34 (1984) no. 4, p. 135-150

For some time it has been known that there exist continuous Helson curves in R 2 . This result, which is related to Lusin’s rearrangement problem, had been proved first by Kahane in 1968 with the aid of Baire category arguments. Later McGehee and Woodward extended this result, giving a concrete construction of a Helson k-manifold in R nk for n≥k+1. We present a construction of a Helson 2-manifold in R 3 . With modification, our method should even suffice to prove that there are Helson hypersurfaces in any R n .

On sait depuis quelque temps que des courbes continues de Helson existent dans r 2 . Kahane a démontré ce résultat, qui est lié au problème de réarrangement de Lusin, en 1968 en utilisant des arguments de catégories de Baire. Plus tard McGehee et Woodward ont étendu ce résultat en donnant une construction concrète d’une variété de Helson à k-dimensions dans R nk pour n≥k+1. Nous présentons une construction d’une variété de Helson à deux dimensions dans R 3 . Avec quelques modifications notre méthode devrait même permettre de prouver l’existence d’hypersurfaces de Helson dans R n pour tout n.

@article{AIF_1984__34_4_135_0,
     author = {M\"uller, Detlef},
     title = {A continuous Helson surface in ${\bf R}^3$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {34},
     number = {4},
     year = {1984},
     pages = {135-150},
     doi = {10.5802/aif.991},
     zbl = {0538.43003},
     mrnumber = {86g:43010},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1984__34_4_135_0}
}
Müller, Detlef. A continuous Helson surface in ${\bf R}^3$. Annales de l'Institut Fourier, Volume 34 (1984) no. 4, pp. 135-150. doi : 10.5802/aif.991. http://www.numdam.org/item/AIF_1984__34_4_135_0/

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