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

We investigate choice principles in the Weihrauch lattice for finite sets on the one hand, and convex sets on the other hand. Increasing cardinality and increasing dimension both correspond to increasing Weihrauch degrees. Moreover, we…

We show that Blokhuis' quadratic upper bound for two-distance sets is sharp over finite fields in almost all dimensions. Our construction complements Lison\v{e}k's higher-dimensional maximal constructions that were carried out in Lorentz…

We study the projections in vector spaces over finite fields. We prove finite fields analogues of the bounds on the dimensions of the exceptional sets for Euclidean projection mapping. We provide examples which do not have exceptional…

The point sets of all known optimal rectilinear drawings of $K_n$ share an unmistakeable clustering property, the so--called {\em 3--decomposability}. It is widely believed that the underlying point sets of all optimal rectilinear drawings…

Consider a domain D in R^3 which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f : M --> D. Moreover, if D is smooth and bounded, then…

We present a streamlined and simplified proof of the Kakeya set conjecture in $\mathbb{R}^3$.

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…

Artin's conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin's conjecture over fields of characteristic p>3. This implies Tate's conjecture for K3 surfaces over finite…

We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K \subseteq \mathbb{R}^2$ an $(s,t)$-Kakeya set if there exists…

It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in $\mathbb{P}^4 (\mathbb{F}_{q^2})$ has at most $d(q^5+q^2) + q^3 + 1$ points in common with a threefold of degree $d$ defined over $\mathbb{F}_{q^2}$. He…

We provide new upper bounds on N_q(g), the maximum number of rational points on a smooth absolutely irreducible genus-g curve over F_q, for many values of q and g. Among other results, we find that N_4(7) = 21 and N_8(5) = 29, and we show…

We study a problem of minimal surfaces with free boundary written in the form of a non convex minimization problem. Our aim is to characterize optimal solutions by finding a suitable calibration field. A natural upper bound of the infimum…

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for…

We discuss the notion of essential dimension of a finite group and explain its relation with birational algebraic geometry. We show how this leads to a (partial) classification of simple finite groups of essential dimension less than or…

Various effective field theories in four dimensions are shown to have exact non-trivial solutions in the limit as the number $N$ of fields of some type becomes large. These include extended versions of the U(N) Gross-Neveu model, the…

Let $\alpha(\mathbb{F}_q^{d},p)$ be the maximum possible size of a point set in general position in the $p$-random subset of $\mathbb{F}_q^d$. In this note, we determine the order of magnitude of $\alpha(\mathbb{F}_q^{3},p)$ up to a…

Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let $K$ be an algebraically closed field of characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$ endowed with…

We consider the maximal number of arbitrary points in a special fibre that can be simultaneously approached by points in one sequence of general fibres. Several results about this topological invariant and their applications describe the…