Related papers: Kakeya sets from lines in $SL_2$

We give conditions for $k$-point configuration sets of thin sets to have nonempty interior, applicable to a wide variety of configurations. This is a continuation of our earlier work \cite{GIT19} on 2-point configurations, extending a…

We characterize the permutations of $\mathbb{F}_q$ whose graph minimizes the number of collinear triples and describe the lexicographically-least one, affirming a conjecture of Cooper-Solymosi. This question is closely connected to…

Motivated by a question of Erd\H{o}s on blocking sets in a projective plane that intersect every line only a few times, several authors have used unions of algebraic curves to construct such sets in $\mathbb{P}^2(\mathbb{F}_q)$. In this…

The distance set $\Delta(E)$ of a set $E$ consists of all non-negative numbers that represent distances between pairs of points in $E$. This paper studies sparse (less than full-dimensional) Borel sets in $\mathbb R^d$, $d \geq 2$ with a…

In this paper we study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace $\mathbf{K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that…

By using Ikeda's theory for a compatible family of Eisenstein series, we explicitly construct Ikeda type lifts on the special orthogonal group $G={\rm SO}(3,n+1)$ over $\mathbb{Q}$ with $n\ge 3$ which splits everywhere at finite places. Our…

Let $X$ be an integral projective variety of codimension two, degree $d$ and dimension $r$ and $Y$ be its general hyperplane section. The problem of lifting generators of minimal degree $\sigma$ from the homogeneous ideal of $Y$ to the…

This article is a study guide for ``On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane" by Orponen and Shmerkin. We begin by introducing Furstenberg set problem and exceptional set of projections and…

We introduce a new family of random compact metric spaces $\mathcal{S}_\alpha$ for $\alpha\in(1,2)$, which we call stable shredded spheres. They are constructed from excursions of $\alpha$-stable L\'evy processes on $[0,1]$ possessing no…

We construct, as hypersurfaces in toric varieties, Calabi-Yau manifolds corresponding to F-theory vacua dual to E8*E8 heterotic strings compactified to six dimensions on K3 surfaces with non-semisimple gauge backgrounds. These vacua were…

In section 1 we consider a 3-tuple $S=(|S|,\preccurlyeq,E)$ where $|S|$ is a finite set, $\preccurlyeq$ a partial ordering on $|S|,$ and $E$ a set of unordered pairs of distinct members of $|S|,$ and study, as a function of $n\geq 0,$ the…

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of…

Let $E \subseteq R^n$ be a closed set of Hausdorff dimension $\alpha$. For $m \geq n$, let $\{B_1,\ldots,B_k\}$ be $n \times (m-n)$ matrices. We prove that if the system of matrices $B_j$ is non-degenerate in a suitable sense, $\alpha$ is…

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…

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. It is shown in [Dai, Rao and Zhang, Space-filling curves of self-similar sets (II): Edge-to-trail substitution…

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these…

We suggest a new algorithm to estimate representations of compact Lie groups from finite samples of their orbits. Different from other reported techniques, our method allows the retrieval of the precise representation type as a direct sum…

We study the Macroscopic Hausdorff dimension of the upper and lower level sets of the Airy processes, following the general method developed in Khoshnevisan et al. \cite{KKX17}. For the Airy$_1$ process, the approach to macroscopic…