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Growth conditions for thin sets in Vilenkin groups of bounded order

Author: D. J. Grubb
Journal: Proc. Amer. Math. Soc. 119 (1993), 567-571
MSC: Primary 43A46; Secondary 42C25
MathSciNet review: 1151811
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Abstract: Let $ G$ be a Vilenkin group of bounded order and $ {H_n}$ a sequence of clopen subgroups of $ G$ forming a base at the identity. If $ E$ is a subset of $ G$, let $ {N_n}(E)$ denote the number of cosets of $ {H_n}$ which intersect $ E$. If

$\displaystyle \underline {\lim } \frac{{{N_n}(E)}} {{\log [G:{H_n}]}} < \infty ,$

then $ E$ is a U-set in the group $ G$. It is also shown that for $ G$ satisfying a growth condition and $ \varphi (n) \to \infty $, there is an M-set, $ E$, with

$\displaystyle {N_n}(E) = O(\varphi (n)\,\log [G:{H_n}]).$

References [Enhancements On Off] (What's this?)

  • [1] D. J. Grubb, $ U$-sets in compact, 0-dimensional, metric groups, Canad. Math. Bull. 32 (1989), 149-155. MR 1006739 (90g:42052)
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  • [3] J. P. Kahane, A metric condition for a closed circular set to be a set of uniqueness, J. Approx. Theory 2 (1969), 233-246. MR 0247363 (40:631)
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Article copyright: © Copyright 1993 American Mathematical Society

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