Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1993-1151811-5
MathSciNet review: 1151811
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)
  • [2] -, Dirichlet sets in Vilenkin groups, Acta Math. Hungar. (to appear). MR 1237000 (94j:43005)
  • [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)
  • [4] Robert Kaufman, Kronecker sets and metric properties of $ {M_0}$-sets, Proc. Amer. Math. Soc. 36 (1972), 519-524. MR 0310540 (46:9638)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 43A46, 42C25

Retrieve articles in all journals with MSC: 43A46, 42C25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1151811-5
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society