Sets with Large Intersection Properties

Abstract

For 0 < s < n let ^" be the class of G r subsets of W such that Fe<$° if {\£.J t (F) h a s Hausdorff dimension at least s for all sequences of similarity transformations {fJffL y We show that <&' is closed under countable intersections and under bi-Lipschitz functions, and thus is the maximal class of G^-sets of Hausdorff dimension at least s that is closed under countable intersection and similarities. We also show that sets in W must have packing dimension n. Many examples of ^'-sets occur in Diophantine approximation.here \U\ is the diameter of U. The s-dimensional Hausdorff (outer) measure is defined by 2tf\F) = lim^o^CF) and is a Borel measure on subsets of R n . The Hausdorff dimension of F is given by dim H F= inf{s:jr s (F) < oo} = sup {s:^s(F) = 0}; see [9] for further details of these concepts.

On the dimensions of sections for the graph-directed sets

“…(3) If A ⊂ R n is a Borel set and 0 ≤ t < dim H A, then there is a compact set B ⊂ A satisfying 0 < H t (B) < ∞ (e.g., see [Fa2]). …”

Ubiquity and large intersections properties under digit frequencies constraints

“…We also show that, under suitable assumptions, such an intersection provides a new type of limsup set enjoying the remarkable property introduced by Falconer in [16,17], called "large intersection property", and defined as follows. Definition 1·4.…”

“…Definition 1·4. [17] For 0 < α ≤ 1, the class G α of sets with large intersection consists in the G δ -subsets Ω ⊂ R such that for all sequences of similarity transformations…”

Approximation properties of -expansions II

“…Persson and Reeve restrict to the case where Ψ(n) = 2 −nα for some α ∈ (1, ∞), so by the Borel-Cantelli lemma K β (2 −nα ) is of zero Lebesgue measure for any β ∈ (1, 2). Motivated by Falconer [7] they studied the intersection properties of K β (Ψ). In [7] Falconer defined G s to be the set of A ⊆ R, which have the property that for any countable collection of similarities {f j } ∞ j=1 , we have…”

“…Motivated by Falconer [7] they studied the intersection properties of K β (Ψ). In [7] Falconer defined G s to be the set of A ⊆ R, which have the property that for any countable collection of similarities {f j } ∞ j=1 , we have…”