Weakly almost periodic functions and thin sets in discrete groups

Chou, Ching

doi:10.1090/S0002-9947-1990-0984855-6

Weakly almost periodic functions and thin sets in discrete groups
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by Ching Chou

Trans. Amer. Math. Soc. 321 (1990), 333-346

DOI: https://doi.org/10.1090/S0002-9947-1990-0984855-6

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Abstract:

A subset $E$ of an infinite discrete group $G$ is called (i) an ${R_W}$-set if any bounded function on $G$ supported by $E$ is weakly almost periodic, (ii) a weak $p$-Sidon set $(1 \leq p < 2)$ if on ${l^1}(E)$ the ${l^p}$-norm is bounded by a constant times the maximal ${C^*}$-norm of ${l^1}(G)$, (iii) a $T$-set if $xE \cap E$ and $Ex \cap E$ are finite whenever $x \ne e$, and (iv) an $FT$-set if it is a finite union of $T$-sets. In this paper, we study relationships among these four classes of thin sets. We show, among other results, that (a) every infinite group $G$ contains an ${R_W}$-set which is not an $FT$-set; (b) countable weak $p$-Sidon sets, $1 \leq p < 4/3$ are $FT$-sets.

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