Nearest neighbour distance and dimension of intensity measure of Poisson point process

Abstract:

We prove that the upper and lower local dimensions of a finite measure $\mu$ are equal to the upper and lower limit of $-\log \alpha / \log \hat {r}_{\alpha \mu }(x)$, where $\hat {r}_{\alpha \mu }(x)$ is the mean distance to the closest point for the Poisson point processes with intensity measure $\alpha \mu$. Moreover the upper local dimension of $\mu$ is a.e. bounded from above by the limit superior of $-\log \alpha / \log \hat {r}_{\alpha \mu }$, where $\hat {r}_{\alpha \mu }$ denotes the expected nearest-neighbour distance.