Quantitative metric density and applications

We begin by discussing the Lebesgue Density Theorem. Then, we show why a possible statement of a “Quantitative Lebesgue Density Theorem” is false, by providing a counterexample using dyadic intervals. Finally, we prove a “Quantitative Metric Density Theorem” on doubling metric measure spaces and provide applications. The first application uses the Quantitative Metric Density and the Hausdorff distance to show that if one moves a closed ball, there won’t be many large gaps that the balls has to skip over. The second application is using the Quantitative Metric Density and applying it on Dyadic Cubes in Rn to show that one may form a path between two points by making small jumps onto points in a measurable set.