Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 141-165

AVERAGE BOX DIMENSIONS OF TYPICAL COMPACT SETS

Lars Olsen

University of St. Andrews, Department of Mathematics
St. Andrews, Fife KY16 9SS, Scotland; lo 'at' st-and.ac.uk

Abstract. Let X be a metric space and write K(X) for the family of non-empty compact subsets of X equipped with the Hausdorff metric. The lower and upper box dimensions, denoted by \underline dim_B(E) and \overline dim_B(E), of a subset E of X are defined by

\underline dim_B(E) = liminf_r&rarr0 log N_r(E)/–log r, \overline dim_B(E) = limsup_r→0 log N_r(E)/–log r,

where N_r(E) is the smallest number of closed balls with centres in E and radii equal to r that are needed to cover E. In the 1980's, Gruber proved that the box counting function

(*) log N_r(C)/–log r

of a typical compact set C ∈ K(X) diverges in the worst possible way as r → 0. For example, Gruber proved that \underline dim_B(C) = 0 and \overline dim_B(E) = N for a typical C ∈ K(R^N).

In this paper we prove that the box counting function (*) of a typical compact set C ∈ K(X) is spectacularly more irregular than suggested by Gruber's result. In particular, we show the following surprising result: not only is the box counting function (*) of a typical compact set C ∈ K(X) divergent as r → 0, but it is so irregular that it remains spectacularly divergent as r → 0 even after being "averaged" or "smoothened out" using powerful averaging methods including, for example, all higher order Hölder and Cesaro averages. As an application of our results we obtain strengthened versions of Gruber's result.

2010 Mathematics Subject Classification: Primary 28A78, 28A80.

Key words: Box dimension, compact set, Hölder mean, Cesaro mean, Baire category.

Reference to this article: L. Olsen: Average box dimensions of typical compact sets. Ann. Acad. Sci. Fenn. Math. 44 (2019), 141-165.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4406