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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weakly almost periodic functions and thin sets in discrete groups


Author: Ching Chou
Journal: Trans. Amer. Math. Soc. 321 (1990), 333-346
MSC: Primary 43A46; Secondary 43A07, 43A30, 43A60
DOI: https://doi.org/10.1090/S0002-9947-1990-0984855-6
MathSciNet review: 984855
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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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Keywords: Discrete groups, weakly almost periodic functions, infinite triangles, large squares, large <IMG WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$k$">-cubes, wide strips, <IMG WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img6.gif" ALT="$T$">-sets, weak <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$p$">-Sidon sets, <IMG WIDTH="39" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img30.gif" ALT="${R_W}$">-sets
Article copyright: © Copyright 1990 American Mathematical Society