Mathematische Annalen

, Volume 331, Issue 3, pp 557–576

# Average homogeneity and dimensions of measures

• Esa Järvenpää
• Maarit Järvenpää
Article

## Abstract.

We introduce the concept of average homogeneity of a measure by comparing the measure to the uniform distribution in a relatively simple way. This leads to a very general notion which may be regarded as an inverse of porosity. In this paper the emphasis is given to relations between homogeneity and dimensions of measures. First we consider the effect of homogeneity on dimensions by proving an upper bound to the Hausdorff dimension as a function of homogeneity and its order. The opposite question of how dimensions effect homogeneity is solved by giving an upper bound to homogeneity in terms of upper packing dimension. We also illustrate by examples that all our results are the best possible ones.

## Keywords

Porosity Uniform Distribution Hausdorff Dimension General Notion Dimension Effect
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