Laura Venieri
We consider (bounded) Besicovitch sets in the Heisenberg group and prove that $L^p$ estimates for the Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.
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Hausdorff dimension of Besicovitch sets of Cantor graphs
本文研究平面中 Cantor 图的 Besicovitch 集的 Hausdorff 维数下界，证明对一大类 Cantor 集及其图，维数至少为 $\min(2-s^2, 1/s)$，其中 $s$ 为 Cantor 集的维数。
The Besicovitch covering property (BCP) is known to be one of the fundamental tools in measure theory, and more generally, a usefull property for numerous purposes in analysis and geometry. We prove both sufficient and necessary criteria for the validity of BCP in the first Heisenberg group equipped with a homogeneous distance. Beyond recovering all previously known results about the validity or non validity of BCP in this setting, we get simple descriptions of new large classes of homogeneous distances satisfying BCP. We also obtain a full characterization of rotationally invariant distances …
Our main result is a positive answer to the question whether one can find homogeneous distances on the Heisenberg groups that have the Besicovitch Covering Property (BCP). This property is well known to be one of the fundamental tools of measure theory, with strong connections with the theory of differentiation of measures. We prove that BCP is satisfied by the homogeneous distances whose unit ball centered at the origin coincides with an Euclidean ball. Such homogeneous distances do exist on any Carnot group by a result of Hebisch and Sikora. In the Heisenberg groups, they are related to the …
We prove that the Besicovitch Covering Property (BCP) does not hold for some classes of homogeneous quasi-distances on Carnot groups of step 3 and higher. As a special case we get that, in Carnot groups of step 3 and higher, BCP is not satisfied for those homogeneous distances whose unit ball centered at the origin coincides with a Euclidean ball centered at the origin. This result comes in constrast with the case of the Heisenberg groups where such distances satisfy BCP.
An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group.
We pose the following conjecture: (*) If A is the union of line segments in R^n, and B is the union of the corresponding full lines then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in R^n has Hausdorff dimension at least n-1 and (upper) Minkowski dimension n. We also prove that conjecture (*) holds if the Hausdorff dimension of B is at most 2, so in particular it holds in the plane.
Besicovitch's 1/2 problem and linear programming
该论文改进 Besicovitch 1/2 猜想的最佳界限，证明下密度阈值可提升至 7/10，并提出变分问题族以产生更多界限。
We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the $\{xoy\}$-plane is a Kakeya set with Heisenberg Hausdorff dimension 3.
A short note on the dimension of the CC-geodesic Kakeya sets in the first Heisenberg group
该研究针对第一Heisenberg群的CC测地线Kakeya集，修正了Kakeya猜想的自然预测，证明其精确下界为4，构造了满维零测集及不同曲率的低维零测Kakeya集，完善了相关维数理论。
Packing dimension of vertical projections in the Heisenberg group
研究Heisenberg群中垂直投影的打包维度，利用变量系数局部平滑不等式改进了Hausdorff维度的下界。
This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of $\mathbb{R}^{2n}$, as well as dimension of intersections of sets with isotropic planes. It is shown that if $A$ and $B$ are Borel subsets of $\mathbb{R}^{2n}$ of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of $A$ and $B$ under orthogonal projections onto these planes have positive Hausdorff $m$-measure. In addition, if $A$ is a measurable set of Hausdorff dimension greater than $m$, then there is a set $B\subset\mathbb{R}^{2n}…
We show that there is a sequence of subsets of each discrete Heisenberg group for which the non-singular ergodic theorem holds. The sequence depends only on the group; it works for any of its non-singular actions. To do this we use a metric which was recently shown by Le Donne and Rigot to have the Besicovitch covering property and then apply an adaptation of Hochman's proof of the multiparameter non-singular ergodic theorem. An exposition of how one proves non-singular ergodic theorems of this type is also included, along with a new proof for one of the key steps.
The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-Caratheodory distances and prove that this also fails for these distances as soon as there are length minimizing curves that stop to be minimizing in finite time. Next we study some connections with the comparison between Hausdorff and spherical Hausdorff me…
This paper establishes connections between the group-Fourier transform and the geometry of measures in the Heisenberg group. Firstly, it is shown that if the Fourier transform of a compactly supported, finite, Radon measure is square integrable, then the measure must have a square integrable density. If it's Fourier transform is integrable, the the measure must have a continuous density. In addition, an alternative formulation of the Fourier transform on the Heisenberg group is used to show that energies of measures can be computed via integrals on an appropriate frequency space. This in turns…
We show that if $A$ is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of $A$ is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff $3$-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.
Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group
本文通过初等变分法证明亚洛伦兹海森堡群的极大测地线为双曲线提升，确定其洛伦兹豪斯多夫维数为4，并揭示该群不满足任何参数的类时曲率维数条件与测度收缩性质。
A Euclidean Fourier-analytic approach to vertical projections in the Heisenberg group
本文利用欧几里得傅里叶分析、Basset积分公式和第二类修正Bessel函数，改进了第一海森堡群中垂直投影下Hausdorff维数的几乎处处下界，并回答了Fässler和Hovila的问题。
On uniform measures in the Heisenberg group
本文研究Heisenberg群中一致测度的分类，完整刻画1维和2维情形，并在支集为竖直直纹曲面时证明3维猜想，同时给出外蕴球测度渐近公式。
Well and badly approximable sets, and rapid winning
该研究引入Ψ-快速博弈确定τ-逼近数集与非齐次恶劣逼近数集交集的豪斯多夫维数，得到了τ≥1时该交集的精确Jarník–Besicovitch维数。
We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive analysis in a new case of a map $F\in C^1_H(\Bbb{H}^1, \Bbb{R}^2)$ with surjective horizontal differential (an analogue of the classical implicit function theorem). Among other results, we show that a level set of such map is locally a simple curve of Hausdorff sub-Riemannian dimension 2, but, surprisingly, in general its two-dimensional Hausdorff measure can be …
We study the behavior of Sobolev mappings defined on the Heisenberg groups with respect to a foliation by left cosets of a horizontal homogeneous subgroup. We quantitatively estimate, in terms of Euclidean Hausdorff dimension, the size of the set of cosets that are mapped onto sets of high dimension. The proof of our main result combines ideas of Gehring and Mostow about the absolute continuity of quasiconformal mappings with Mattila's projection and slicing machinery.
Kakeya maximal inequality in the Heisenberg group
本文在配备 Korányi 距离的海森堡群中定义了 Kakeya 极大函数，利用 Wolff 圆形极大函数定理的变体证明了其 $L^3$ 范数不等式，并由此恢复了海森堡 Kakeya 集 Hausdorff 维度的精确下界。
We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $\mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to the Euclidean metric and also the natural quotient metric which we show behaves like the Euclidean metric in this context. Our bounds are sharp in a large part of the dimension range, and we give conjectural sharp lower bounds for the remaining range. Our approach also lets us improve the known almost sure lower bound for the standard family of vertical proje…
Elementary sub-Riemannian geometry on the Heisenberg group H(n) provides a compact picture of symplectic geometry. Any Hamiltonian diffeomorphism on $R^{2n}$ lifts to a volume preserving bi-Lipschitz homeomorphisms of H(n), with the use of its generating function. Any curve of a flow of such homeomorphisms deviates from horizontality by the Hamiltonian of the flow. From the metric point of view this means that any such curve has Hausdorff dimension 2 and the $\mathcal{H}^{2}$ (area) density equal to the Hamiltonian. The non-degeneracy of the Hofer distance is a direct consequence of this fact.
Vertical curves and vertical fibers in the Heisenberg group
研究海森堡群中垂直曲线及相关纤维，通过证明连通垂直曲线性质、表明其与特定内在Lipschitz图交集类一致且维数可变，以及研究相关光滑函数纤维测度，揭示了海森堡群中这些对象的独特性质。
A note on limsup sets of annuli
该研究探讨$\mathbb R^{n}$中以有理点为中心的无穷极大范数环带的上极限集，给出其豪斯多夫维数的Jarník-Besicovitch型定理，结合Cassel缩放引理与推广的质量转移原理，发现特定半径和厚度条件下维数趋近于$n-1$，还考虑了其他环带形式。
We initiate the study of an intrinsic notion of Diophantine approximation on a rational Carnot group $G$. If $G$ has Hausdorff dimension $Q$, we show that its Diophantine exponent is equal to $(Q+1)/Q$, generalizing the case $G=\mathbb R^n$. We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group $\mathbb H^n$, distinguishing between two notions of Diophantine approximation by rational points in $\mathbb H^n$: Carnot Diophantine approximation and Siegel Diophantine approximation. After computing the Siegel Diophantin…
In Carnot groups of step 2 we consider sets having maximal or minimal possible homogeneous Hausdorff dimension compared to their Euclidean one: in the first case we prove that they must be in a sense vertical, that is a large part of these sets can lie off horizontal planes through their points, and in the latter case close to horizontal. The examples that had been used to prove the sharpness of the sub-Riemannian versus Euclidean dimension comparison intuitively satisfied these geometric properties, which we here quantify and prove to be necessary. We also show the sharpness of our statements…
Unrectifiable surfaces with intrinsic regularity in Carnot groups
本文在非交换Carnot群中构造了满足0≤α<1的C^{1,α}_H不可求积曲面，在步长为2的Carnot群中得到其最大欧氏Hausdorff维数，相关结果扩展了前人在Heisenberg群中的工作。
Vertical projections in the Heisenberg group via cinematic functions and point-plate incidences
该研究针对第一海森堡群中的垂直投影问题，分别借助cinematic函数方法、引入水平线与圆锥线的点线对偶转化为点-板关联问题，拓展了博雷尔集豪斯多夫维数相关投影结论的适用范围。
In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results. In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying c…
This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a $C^{\infty}$ hypersurface $S$ without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure $\mat…
We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\mathrm{MCP}(K,N)$ if and only if $K \leq 0$ and $N \geq k+3$. This generalizes the well known result by Juillet for the Heisenberg group $\mathbb{H}_{k+1}$ to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number $k+3$ coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least $N$ such that the $\mathrm{MC…
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