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

Hajela, D. J.

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

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

– there is a $\Lambda$ (2k) set which is not $\Lambda (2k+\epsilon )$ in $\mathbf{Z}\left(2\right)\oplus \mathbf{Z}\left(2\right)\oplus \cdots$ for all $2\le k$, $k\in \mathbf{N}$ and $\epsilon >0$, and in $\mathbf{Z}\left(p\right)\oplus \mathbf{Z}\left(p\right)\oplus \cdots$ ($p$ a prime, $p>2$) for $2\le k<p$, $k\in \mathbf{N}$ and $\epsilon >0$ (answering a question in J. Lopez and K. Ross, Marcel Dekker, 1975),

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

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

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

– il existe un ensemble $\Lambda \left(2k\right)$ qui n’est pas $\Lambda (2k+\epsilon )$ en $\mathbf{Z}\left(2\right)\oplus \mathbf{Z}\left(2\right)\oplus \cdots$ quel que soit $2\le k$, $k\in \mathbf{N}$ et $\epsilon >0$ ainsi que dans $\mathbf{Z}\left(p\right)\oplus \mathbf{Z}\left(p\right)\oplus \cdots$ ($p$ étant un nombre premier, $p>2$) pour $2\le k<p$, $k\in \mathbf{N}$ et $\epsilon >0$ (pour répondre à une question posée dans J. Lopez and K. Ross, Marcel Dekker, 1975),

– il existe un ensemble $\Lambda \left(2k\right)$ qui n’est pas $\Lambda (4k-4+\epsilon )$ dans $\mathbf{Z}\left(p^{\infty }\right)$ pour $2\le k$, $k\in \mathbf{N}$ et tout $\epsilon >0$.

Il est également démontré que des suites aléatoires illimitées en entiers sont $\Lambda \left(2k\right)$ et non pas $\Lambda (2k+\epsilon )$ pour $2\le k$, $k\in \mathbf{N}$ et $\epsilon >0$.

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