Abstract
We investigate Minkowski, or box-counting, dimension of universal differentiability
sets of Lipschitz functions. Whilst existing results concern Lebesgue measure and Hausdorff dimension of these fractal sets, the Minkowski dimension is stronger than Hausdorff, and we demonstrate that the lower bound one on Minkowski dimension is tight for any Euclidean space. Spaces other than the real line allow for a further refinement of the bound: the 1-Hausdorff measure of such sets must be infinite.
sets of Lipschitz functions. Whilst existing results concern Lebesgue measure and Hausdorff dimension of these fractal sets, the Minkowski dimension is stronger than Hausdorff, and we demonstrate that the lower bound one on Minkowski dimension is tight for any Euclidean space. Spaces other than the real line allow for a further refinement of the bound: the 1-Hausdorff measure of such sets must be infinite.
| Original language | English |
|---|---|
| Pages (from-to) | 613-636 |
| Number of pages | 24 |
| Journal | Michigan Mathematical Journal |
| Volume | 65 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2016 |