Related papers: Discretized Radial Projections in $\mathbb{R}^d$

We establish a $p$-adic analogue of a recent significant result of Ren-Wang (arXiv:2308.08819) on Furstenberg sets in the Euclidean plane. Building on the $p$-adic version of the high-low method from Chu (arXiv:2510.20104), we analyze…

Pirashvili's Dold-Kan type theorem for finite pointed sets follows from the identification in terms of surjections of the morphisms between the tensor powers of a functor playing the role of the augmentation ideal; these functors are…

The article proves an assertion analogous to the Littlewood-Paley theorem for the orthoprojectors onto mutually orthogonal subspaces of piecewise polynomial functions on the cube $ I^d. $ This assertion provides an upper estimate for the…

By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(\delta,\sigma)$-set, with $|A| = \delta^{-\sigma},$ then…

In this paper we prove that for any definable subset $X\subset \mathbb{R}^{n}$ in a polynomially bounded o-minimal structure, with $dim(X)<n$, there is a finite set of regular projections (in the sense of Mostowski ). We give also a weak…

In this paper we prove two general results related to Marstrand's projection theorem in a quite general formulation over separable metric spaces under a suitable transversality hypothesis (the "projections" are in principle only measurable)…

The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study…

The Wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits. To overcome this obstacle, for probability measures supported on finitely many points, we derive…

We study extensions and generalizations of the Schmidt Subspace Theorem in various settings. In particular, we prove results for algebraic points of bounded degree, giving a sharp version of Schmidt's theorem for quadratic points in the…

According to the von Neumann-Halperin and Lapidus theorems, in a Hilbert space the iterates of products or, respectively, of convex combinations of orthoprojections are strongly convergent. We extend these results to the iterates of convex…

We prove discrete restriction estimates for a broad class of hypersurfaces and varieties of intermediate codimension. For our result about hypersurfaces, we use Bourgain's arithmetic version of the Tomas--Stein method and Magyar's…

In this note we study some basic properties of general fractional derivatives induced by weighted Bergman kernels. As an application we demonstrate a method for generating pre-images of analytic functions under weighted Bergman projections.…

Let $\Delta_n$ and $Q_n$ denote the regular $n$-simplex of side length $\sqrt{2}$ embedded in $\mathbb{R}^{n+1}$ and the volume one cube in $\mathbb{R}^n$, respectively. We derive a closed-form formula for the hyperplane volume projections…

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a…

In this paper, we prove an extension theorem for spheres of square radii in $\mathbb{F}_q^d$, which improves a result obtained by Iosevich and Koh (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a…

For $2 < p < p_0 \simeq 26.265$, the hyperplane section of the $l_p^n$-unit ball $B_p^n$ perpendicular to a^(n) = 1/sqrt(n) (1, ... ,1) for large $n$ has larger volume than the one orthogonal to a^(2) = 1/sqrt(2) (1,1,0, ...,0), as shown by…

For $e \in S^{2}$, the unit sphere in $\mathbb{R}^3$, let $\pi_{e}$ be the orthogonal projection to $e^{\perp} \subset \mathbb{R}^{3}$, and let $W \subset \mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. We prove that if $K…

We study random waves on smooth, compact, Riemannian manifolds under the spherical ensemble. Our first main result shows that there is a positive universal limit for the critical radius of a specific deterministic embedding, defined via the…

In this paper we prove using quite elementary methods, with a combinatorial nature, two general results related to Marstrand's projection theorem in a quite general formulation over metric spaces under a suitable transversality condition…

The Rolling Ball Theorem asserts that given a convex body K in Euclidean space and having a smooth surface bd(K) with all principal curvatures not exceeding c>0 at all boundary points, K necessarily has the property that to each boundary…