Related papers: Finite field Kakeya and Nikodym sets in three dime…

We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to…

Let $A, B$, be finite subsets of an abelian group, and let $G \subset A \times B$ be such that $# A, # B, # \{a+b: (a,b) \in G \} \leq N$. We consider the question of estimating the quantity $# \{a-b: (a,b) \in G \}$. Recently Bourgain…

We improve the well-known Szemer\'edi-Trotter incidence bound for proper 3--dimensional point sets (defined appropriately)

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension L/K such that X has ordinary reduction at every non-archimedean place of L outside a density zero set of places.

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors…

We give upper bounds on the number of exceptional radial projections of arbitrary subsets of vector spaces over finite fields. Our bounds do not depend on the dimension of the ambient space. Let $\mathbb{F}_q^d$ be the $d$-dimensional…

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least…

It is a theorem of Ribet that an abelian variety defined over a number field $K$ has only finitely many torsion points with values in the maximal cyclotomic extension field $K^{\mathrm{cyc}}$ of $K$. Recently, R\"ossler and Szamuely…

We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines of the form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We show that if $\mathcal{L}$ is a Kakeya set of $SL(2)$ lines, then the union $\cup…

We show that if a collection of lines in a vector space over a finite field has "dimension" at least 2(d-1) + beta, then its union has "dimension" at least d + beta. This is the sharp estimate of its type when no structural assumptions are…

We prove that a geometrically integral smooth 3-fold $X$ with nef anti-canonical class and negative Kodaira dimension over a finite field $\mathbb{F}_q$ of characteristic $p>5$ and cardinality $q=p^e > 19$ has a rational point.…

Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued…

We prove that the Kakeya maximal conjecture is equivalent to the $\Omega$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{\'e} proved that the Kakeya conjecture is equivalent to the $\Omega$-Kakeya…

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres…

We give multiple descriptions of a topological universe of finitary sets, which can be seen as a natural limit completion of the hereditarily finite sets. This universe is characterized as a metric completion of the hereditarily finite…

In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of…

We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order…

We obtain new bounds for the Kakeya maximal conjecture in most dimensions $n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or $9$. For this we consider Guth and Zahl's strengthened formulation of the maximal…

We construct K3 surfaces over number fields that have good reduction everywhere. These do not exists over the rational numbers, by results of Abrashkin and Fontaine. Our surfaces exist for three quadratic number fields, and an infinite…

Finding the maximum size of a Sidon set in $\mathbb{F}_2^t$ is of research interest for more than 40 years. In order to tackle this problem we recall a one-to-one correspondence between sum-free Sidon sets and linear codes with minimum…