Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

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

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Article copyright: © Copyright 1993 American Mathematical Society