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Regular and ubiquitous systems, and M ∞ s ‐dense sequences
Author(s) -
Rynne Bryan P.
Publication year - 1992
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300014960
Subject(s) - intersection (aeronautics) , mathematics , construct (python library) , sequence (biology) , dimension (graph theory) , object (grammar) , hausdorff space , hausdorff dimension , discrete mathematics , pure mathematics , computer science , artificial intelligence , biology , engineering , genetics , programming language , aerospace engineering
“Regular systems” of numbers in ℝ and “ubiquitous systems” in ℝ k , k ≥ 1, have been used previously to obtain lower bounds for the Hausdorff dimension of various sets in ℝ and ℝ k respectively. Both these concepts make sense for systems of numbers in ℝ, but the definitions of the two types of object are rather different. In this paper it will be shown that, after certain modifications to the definitions, the two concepts are essentially equivalent. We also consider the concept of aM ∞ s ‐dense sequence in ℝ k , which was introduced by Falconer to construct classes of sets having “large intersection”. We will show that ubiquitous systems can be used to construct examples ofM ∞ s ‐dense sequences. This provides a relatively easy means of constructingM ∞ s ‐dense sequences; a direct construction and proof that a sequence isM ∞ s ‐dense is usually rather difficult.