arXiv subjects
Publications and source records attributed to A. Dali Nimer.
In the following paper, we prove a dimension bound on the singular set of a Radon measure assuming its doubling ratio converges uniformly on compact sets. More precisely, we prove that if a Radon measure is $n$-Uniformly Asymptotically Doubling, then $\dim(\mathcal{S}_μ) \leq n-3$, where $\mathcal{S}_μ$ is the singular set of the measure.
The study of uniformly distributed measures was crucial in Preiss' proof of his theorem on rectifiability of measures with positive density. It is known that the support of a uniformly distributed measure is an analytic variety. In this paper, we provide quantitative information on the rectifiability of this variety. Tolsa had already shown that $n$-uniform measures have Big Pieces of Lipschitz Graphs(BPLG) . Here, we prove that a uniformly distributed measure has BPLG locally.
Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. For instance, they were essential in the groundbreaking work of Preiss on the rectifiability of Radon measures. However, relatively little is understood about the structure of general uniform measures. Indeed, the question of whether there exist any non-flat uniform measures beside the one supported on the light cone has been open for 30 years, ever since Kowalski and Preiss classified $n$-uniform measures in $\mathbb{R}^{n+1}$ . In this paper, we answer the question and construct an infinite family of 3-uniform measures in arbitrary codimension. We define a notion of distance symmetry for points and prove that every collection of $2$-spheres whose centers are distance symmetric gives rise to a $3$-uniform measure. We then develop a combinatorial method to systematically produce distance symmetric points. We also classify conical $3$-uniform measures in $\mathbb{R}^{5}$ by proving that they all arise from distance symmetric spheres.
The study of the geometry of $n$-uniform measures in $\mathbb{R}^{d}$ has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely $3$-Hausdorff measure restricted to the Kowalski-Preiss cone. Using this cone one can construct an $n$-uniform measure whose singular set has Hausdorff dimension $n-3$. In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any $n$-uniform measure is at most $n-3$.