Sets with large intersection properties in metric spaces

Large intersection property for limsup sets in metric space

“…Let (X, ρ) be a metric space endowed with doubling measure µ. Denote τ = dim H X, and τ is finite, see [11,Section 13]. In [21], Negreira and Sequeira gave the definition of the classes of G δ sets with large intersection property. Recall that a G δ set is a countable intersection of open sets.…”

“…(1) Negreira and Sequeira [21] showed that the net contents given by different dyadic decomposition are equivalent, and pointed out that the net content is defined by…”

“…In this paper we only consider such measures, hence we use the notation G t (X) instead of G t µ (X). There are several equivalent definitions of G t (X), see [21]. In this paper, we will use the following equivalent definition.…”

“…In 2004, Bugeaud [3] generalized the class for more general gauge functions. Later, Negreira and Sequeira [21] extended the class of sets in Euclidean space with large intersection property to metric space endowed with doubling measure.…”

“…There are many applications of large intersection property. Lots of mathematicians applied this property for estimating Hausdorff dimension from below in the study of Diophantine approximations, and refer to [3], [4], [9], [21] for more details. In 2007, Durand [5] investigated the size and large intersection properties of limsup sets generated by homogeneous ubiquitous systems in R d .…”