Fractional cartesian products of sets
Blei, Ron C.
Annales de l'Institut Fourier, Tome 29 (1979), p. 79-105 / Harvested from Numdam
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Soit E un sous-ensemble du dual Γ d’un groupe compact G. On dit que E est exactement p-Sidon (resp. exactement non-p-Sidon) quand (*)C E (G) ∧ ⊂ℓ r si et seulement si r∈[p,∞] (resp. r∈(p,∞)). On dit que E est exactement Λ β (resp. exactement non-Λ β ) quand E vérifie (**)toute f∈L E 2 (G) est telle que, quel que soit λ>0,

∫G exp (λ|f|2/α<∞

si et seulement si α∈[β,∞) (resp. α∈(β,∞)).

Dans ce travail, pour chaque p∈[1,2) et β∈[1,∞), on construit des ensembles qui sont exactement p-Sidon, exactement non-p-Sidon, exactement Λ β et exactement non-Λ β .

Let E be a subset of a discrete abelian group whose compact dual is G. E is exactly p-Sidon (respectively, exactly non-p-Sidon) when (*)C E (G) ∧ ⊂ℓ r holds if and only if r∈[p,∞] (respectively, r∈(p,∞)). E is said to be exactly Λ β (respectively, exactly non-Λ β ) if E has the property (**)everyf∈L E 2 (G)satisfies∫ G exp (λ|f| 2/α <∞,forallλ>0, if and only if α∈[β,∞) (respectively, α∈(β,∞)).

In this paper, for every p∈[1,2) and β∈[1,∞), we display sets which are exactly p-Sidon, exactly non-p-Sidon, exactly Λ β and exactly non-Λ β .

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     author = {Blei, Ron C.},
     title = {Fractional cartesian products of sets},
     journal = {Annales de l'Institut Fourier},
     volume = {29},
     year = {1979},
     pages = {79-105},
     doi = {10.5802/aif.744},
     mrnumber = {81h:43008},
     zbl = {0381.43003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1979__29_2_79_0}
}

Blei, Ron C. Fractional cartesian products of sets. Annales de l'Institut Fourier, Tome 29 (1979) pp. 79-105. doi : 10.5802/aif.744. http://gdmltest.u-ga.fr/item/AIF_1979__29_2_79_0/

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