mark

Suppose that I follow our definition of box-counts from the text quite closely:

N_{\varepsilon}(E) is the smallest number of \varepsilon-mesh cubes whose union contains E.

My wife (who doesn’t like to miss any detail, however fine) uses the following definition:

N_{\varepsilon}(E) is the number of closed \varepsilon-mesh cubes whose intersection with E is non-empty.

- Suppose that my wife and I both compute N_{1/2}(S), where S is the closed unit square in \mathbb R^2. What values do we each obtain?
- Show that, in general, my wife’s count might exceed mine by at most the factor five.
- Show that, regardless, we both obtain the same value for the box-counting dimension of a set in the plane.