Proposition
Let
be any non-empty bounded subset. The following quantities are same:
- the smallest number of sets of diameter at most
that cover ; - the smallest number of closed balls of radius
that cover ; - the smallest number of cubes of side length
that cover ; - the number of
-mesh cubes that intersect ; - the largest number of disjoint balls of radius
with centres in . We shall denote this number by
.
Box-counting Dimension
Let
be any non-empty bounded subset, then we define lower and upper box-counting dimension of as If they are equal, we call it the box-counting dimension of and denote it by .
e.g. Consider the compact set