Related papers: Metric results on inhomogeneously singular vectors

We view space-filling circle packings as subsets of the boundary of hyperbolic space subject to symmetry conditions based on a discrete group of isometries. This allows for the application of counting methods which admit rigorous upper and…

In this paper we initiate a study of the topological group $PPQI(G,H)$ of pattern-preserving quasi-isometries for $G$ a hyperbolic Poincare duality group and $H$ an infinite quasiconvex subgroup of infinite index in $G$. Suppose $\partial…

We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L^2 complex relative to a suitable metric on the bundle and a complete metric on the…

We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by…

We introduce a novel concept of rank for subsets of finite metric spaces E^n_q (the set of all n-dimensional vectors over an alphabet of size q) equipped with the Hamming distance, where the rank R(A) of a subset A is defined as the number…

In this paper we derive estimates for linear potentials that hold away from thin subsets. And, inspired by the celebrated work of Huber, we verify that, for a subset that is thin at a point, there is always a geodesic that reaches to the…

Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set $\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding…

Motivated by various geometric problems, we study the nodal set of solutions to Dirac equations on manifolds, of general form. We prove that such set has Hausdorff dimension less than or equal to $n-2$, $n$ being the ambient dimension. We…

We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence…

Using the variational principle in parametric geometry of numbers, we compute the Hausdorff and packing dimension of Diophantine sets related to exponents of Diophantine approximation, and their intersections. In particular, we extend a…

The classical Khintchine--Jarn\'ik Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired much modern research in metric Diophantine…

We prove a number of results related to the size and propagation of boundary pluripolar sets, the exceptional sets for the Dirichlet problem for the complex Monge--Amp\`ere equation. We extend Stout's result that peak sets on strictly…

Let $X$ be a $d$-dimensional Gaussian process in $[0,1]$, where the component are independent copies of a scalar Gaussian process $X_0$ on $[0,1]$ with a given general variance function $\gamma^2(r)=\operatorname{Var}\left(X_0(r)\right)$…

Let F be a non-archimedean local field, of characteristic 0. Let V be a finite dimensional vector space over F and q be a non-degenerate quadratic form on V. Denote d the dimension of V and G=SO(d) the special orthogonal group of (V,q). Let…

For a one-parameter degeneration of reduced compact complex analytic spaces of dimension $n$, we prove the invariance of the frontier Hodge numbers $h^{p,q}$ (that is, with $pq(n{-}p)(n{-}q)=0$) for the intersection cohomology of the fibers…

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…

<ENGLISH> Consider a closed, smooth manifold M of nonpositive sectional curvature. Write p:UM-> M for the unit tangent bundle over M and let R_> denote the subset consisting of all vectors of higher rank. This subset is closed and invariant…

We construct subsets of Euclidean space of large Hausdorff dimension and full Minkowski dimension that do not contain nontrivial patterns described by the zero sets of functions. The results are of two types. Given a countable collection of…

We prove the existence of extremal, non-csc, K\"ahler metrics on certain unstable projectivised vector bundles $\P (E) \to M$ over a cscK-manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic K\"ahler classes. In…

Besicovitch showed that if a set is null for the Hausdorff measure associated to a given dimension function, then it is still null for the Hausdorff measure corresponding to a smaller dimension function. We prove that this is not true for…