Some Problems About Thickness In Fractal

Newhouse first proposed the concept of thickness for Cantor sets in R in 1979.He proved that the sum of two Cantor sets in R has interior points,if the product of their thicknesses is greater than or equal to 1.Feng-Wu studied the question of whether the arithmetic sum of compact sets has interior points in higher dimensional space Rd in 2018.They proposed the concept of n-dimensional thickness for compact sets in Rd,by which they obtained a sufficient condition such that the arithmetic sum of finitely many compact sets in Rd has interior points.A Cantor set in R is of positive Newhouse thickness if and only if it is of positive Feng-Wu thickness.We say that a compact set E in Rd is of arithmetic thickness,if some finite sum of itself has interior points.In the present paper,we focus on ndimensional thickness of compact sets in Rd.We gave several intuitive examples for a deeper understanding of thickness.Then we formulated properties of n-dimensional thickness and summarized sufficient conditions such that fractal sets are of arithmetic thickness.Based on these,we studied the relationships of weak tangents,centred microsets,and n-dimensional thickness of compact sets.Especially,we constructed a planar compact set of positive n-dimensional thickness,whose weak tangent at each point is non-degenerated and it has degenerated centred microsets.