Construction techniques for some thin sets in duals of compact abelian groups
Annales de l'Institut Fourier, Volume 36 (1986) no. 3, pp. 137-166.

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(2k+ε) in Z(2)Z(2) for all 2k, kN and ε>0, and in Z(p)Z(p) (p a prime, p>2) for 2k<p, kN 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(4k-4+ε) in Z(p ) for 2k, kN and all ϵ>0.

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

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(2k) qui n’est pas Λ(2k+ε) en Z(2)Z(2) quel que soit 2k, kN et ε>0 ainsi que dans Z(p)Z(p) (p étant un nombre premier, p>2) pour 2k<p, kN et ε>0 (pour répondre à une question posée dans J. Lopez and K. Ross, Marcel Dekker, 1975),

– il existe un ensemble Λ(2k) qui n’est pas Λ(4k-4+ε) dans Z(p ) pour 2k, kN et tout ε>0.

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

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     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},
     volume = {36},
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Hajela, D. J. Construction techniques for some thin sets in duals of compact abelian groups. Annales de l'Institut Fourier, Volume 36 (1986) no. 3, pp. 137-166. doi : 10.5802/aif.1063. http://archive.numdam.org/articles/10.5802/aif.1063/

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