A continuous Helson surface in ūĚźĎ 3
Annales de l'Institut Fourier, Volume 34 (1984) no. 4, pp. 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‚Äôs 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‚Č•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 d√©montr√© ce r√©sultat, qui est li√© au probl√®me de r√©arrangement de Lusin, en 1968 en utilisant des arguments de cat√©gories de Baire. Plus tard McGehee et Woodward ont √©tendu ce r√©sultat en donnant une construction concr√®te d‚Äôune vari√©t√© de Helson √† k-dimensions dans R nk pour n‚Č•k+1. Nous pr√©sentons une construction d‚Äôune vari√©t√© de Helson √† deux dimensions dans R 3 . Avec quelques modifications notre m√©thode devrait m√™me permettre de prouver l‚Äôexistence d‚Äôhypersurfaces 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},
     pages = {135--150},
     publisher = {Imprimerie Durand},
     address = {Chartres},
     volume = {34},
     number = {4},
     year = {1984},
     doi = {10.5802/aif.991},
     zbl = {0538.43003},
     mrnumber = {86g:43010},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.991/}
}
TY  - JOUR
AU  - M√ľller, Detlef
TI  - A continuous Helson surface in ${\bf R}^3$
JO  - Annales de l'Institut Fourier
PY  - 1984
DA  - 1984///
SP  - 135
EP  - 150
VL  - 34
IS  - 4
PB  - Imprimerie Durand
PP  - Chartres
UR  - http://www.numdam.org/articles/10.5802/aif.991/
UR  - https://zbmath.org/?q=an%3A0538.43003
UR  - https://www.ams.org/mathscinet-getitem?mr=86g:43010
UR  - https://doi.org/10.5802/aif.991
DO  - 10.5802/aif.991
LA  - en
ID  - AIF_1984__34_4_135_0
ER  - 
%0 Journal Article
%A M√ľller, Detlef
%T A continuous Helson surface in ${\bf R}^3$
%J Annales de l'Institut Fourier
%D 1984
%P 135-150
%V 34
%N 4
%I Imprimerie Durand
%C Chartres
%U https://doi.org/10.5802/aif.991
%R 10.5802/aif.991
%G en
%F AIF_1984__34_4_135_0
M√ľller, 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/articles/10.5802/aif.991/

[1] C. C. Graham and O. C. Mcgehee, Essays in Commutative Harmonic Analysis, Springer-Verlag, New York, 1979. | MR | Zbl

[2] C. S. Herz, Drury's Lemma and Helson sets, Studia Math., 42 (1972), 207-219. | MR | Zbl

[3] J. P. Kahane, Sur les réarrangements de fonctions de la classe A, Studia Math., 31 (1968), 287-293. | MR | Zbl

[4] O. C. Mcgehee, Helson sets in Tn, in : Conference on Harmonic Analysis, College Park, Maryland, 1971 ; Springer-Verlag, New York, 1972, 229-237. | MR | Zbl

[5] O. C. Mcgehee and G. S. Woodward, Continuous manifolds in Rn that are sets of interpolation for the Fourier algebra, Ark, Mat., 20 (1982), 169-199. | MR | Zbl

[6] W. Rudin, Fourier Analysis on Groups, Wiley, New York, 1962. | MR | Zbl

[7] S. Saeki, On the union of two Helson sets, J. Math. Soc. Japan, 23 (1971), 636-648. | MR | Zbl

[8] N. Th. Varopoulos, Sidon sets in Rn, Math. Scand., 27 (1970), 39-49. | MR | Zbl

[9] N. Th. Varopoulos, Groups of continuous functions in harmonic analysis, Acta Math., 125 (1970), 109-152. | MR | Zbl

Cited by Sources: