Construction techniques for some thin sets in duals of compact abelian groups

Hajela, D. J.

doi:10.5802/aif.1063

Hajela, D. J.

Annales de l'Institut Fourier, Tome 36 (1986) no. 3, pp. 137-166.

Résumé
Abstract

Diverses techniques sont présentées pour la construction d’ensembles $Δ (p)$ que ne sont pas $Λ (p + ε)$ quel que soit $ε > 0$ . Il en résulte essentiellement qu’il existe un ensemble $Λ (4)$ dans le dual de tout groupe abélien compact qui n’est pas $Λ (4 + ε)$ quel que soit $ε > 0$ . Au cours de la démonstration de nouvelles constructions sont données en groupes duaux dans lesquels des constructions d’ensembles $Λ (p)$ et non $Λ (p + ε)$ étaient déjà connues, pour certaines valeurs de $p$ . Les principales nouvelles constructions en groupes duaux sont :

– il existe un ensemble $Λ (2 k)$ qui n’est pas $Λ (2 k + ε)$ en $Z (2) \oplus Z (2) \oplus \dots$ quel que soit $2 \leq k$ , $k \in N$ et $ε > 0$ ainsi que dans $Z (p) \oplus Z (p) \oplus \dots$ ( $p$ étant un nombre premier, $p > 2$ ) pour $2 \leq k < p$ , $k \in N$ et $ε > 0$ (pour répondre à une question posée dans J. Lopez and K. Ross, Marcel Dekker, 1975),

– il existe un ensemble $Λ (2 k)$ qui n’est pas $Λ (4 k - 4 + ε)$ dans $Z (p^{\infty})$ pour $2 \leq k$ , $k \in N$ et tout $ε > 0$ .

Il est également démontré que des suites aléatoires illimitées en entiers sont $Λ (2 k)$ et non pas $Λ (2 k + ε)$ pour $2 \leq k$ , $k \in N$ et $ε > 0$ .

Various techniques are presented for constructing $Λ$ (p) sets which are not $Λ (p + ϵ)$ for all $ϵ > 0$ . The main result is that there is a $Λ$ (4) set in the dual of any compact abelian group which is not $Λ (4 + ϵ)$ for all $ϵ > 0$ . Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $Λ$ (p) not $Λ (p + ϵ)$ sets, for certain values of $p$ . The main new constructions in specific dual groups are:

– there is a $Λ$ (2k) set which is not $Λ (2 k + ε)$ in $Z (2) \oplus Z (2) \oplus \dots$ for all $2 \leq k$ , $k \in N$ and $ε > 0$ , and in $Z (p) \oplus Z (p) \oplus \dots$ ( $p$ a prime, $p > 2$ ) for $2 \leq k < p$ , $k \in N$ and $ε > 0$ (answering a question in J. Lopez and K. Ross, Marcel Dekker, 1975),

– there is a $Λ$ (2k) set which is not $Λ (4 k - 4 + ε)$ in $Z (p^{\infty})$ for $2 \leq k$ , $k \in N$ and all $ϵ > 0 .$

It is also shown that random infinite integer sequences are $Λ$ (2k) but not $Λ (2 k + ϵ)$ for $2 \leq k$ , $k \in N$ and $ϵ > 0$ .

MR Zbl

DOI : 10.5802/aif.1063

@article{AIF_1986__36_3_137_0,
     author = {Hajela, D. J.},
     title = {Construction techniques for some thin sets in duals of compact abelian groups},
     journal = {Annales de l'Institut Fourier},
     pages = {137--166},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {36},
     number = {3},
     year = {1986},
     doi = {10.5802/aif.1063},
     zbl = {0586.43004},
     mrnumber = {88c:43007},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1063/}
}

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Hajela, D. J. Construction techniques for some thin sets in duals of compact abelian groups. Annales de l'Institut Fourier, Tome 36 (1986) no. 3, pp. 137-166. doi : 10.5802/aif.1063. https://aif.centre-mersenne.org/articles/10.5802/aif.1063/

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