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Transactions of the American Mathematical Society

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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0984855-6
Keywords: Discrete groups, weakly almost periodic functions, infinite triangles, large squares, large $ k$-cubes, wide strips, $ T$-sets, weak $ p$-Sidon sets, $ {R_W}$-sets
Article copyright: © Copyright 1990 American Mathematical Society

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