Related papers: On the size of Nikodym sets in finite fields

A Nikodym set $\mathcal{N}\subseteq(\mathbb{Z}/(N\mathbb{Z}))^n$ is a set containing $L\setminus\{x\}$ for every $x\in(\mathbb{Z}/(N\mathbb{Z}))^n$, where $L$ is a line passing through $x$. We prove that if $N$ is square-free, then the size…

A Kakeya set is a subset of F^n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least C_n * q^n, where C_n depends only on n. This improves the…

A set of points $N\subseteq \mathbb{F}_q^d$ is a Nikodym set if, for any $x\in \mathbb{F}_q^d$, there is a line $\ell$ through $x$ such that $\ell\setminus\{x\}\subseteq N$. We conjecture that $|N|=q^d-O_d(q^{d/(d-1)})$ and prove it under…

We give improved lower bounds on the size of Kakeya and Nikodym sets over $\mathbb{F}_q^3$. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in $\mathbb{F}_q^3$, and show that…

Let $\mathbb{F}$ be a finite field consisting of $q$ elements and let $n \geq 1$ be an integer. In this paper, we study the size of local Kakeya sets with respect to subsets of $\mathbb{F}^{n}$ and obtain upper and lower bounds for the…

For a finite field GF(q) a Kakeya set K is a subset of GF(q)^n that contains a line in every direction. This paper derives new upper bounds on the minimum size of Kakeya sets when q is even.

A Sidon set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d$. We improve on the lower bound on the diameter of a Sidon set with $k$ elements: if $k$ is sufficiently large and ${\cal A}$ is a Sidon set with…

For any fixed dimension $d \geq 3$ we construct a Nikodym set in $F_q^d$ of cardinality $q^d - (\frac{d-2}{\log 2} +1+o(1)) q^{d-1} \log q$ in the limit $q \to \infty$, when $q$ is an odd prime power. This improves upon the naive random…

A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and…

A Besicovitch set in AG(n,q) is a set of points containing a line in every direction. The Kakeya problem is to determine the minimal size of such a set. We solve the Kakeya problem in the plane, and substantially improve the known bounds…

For any integer $d \geq 2$ and prime power $q$, we construct unexpectedly large induced matchings in the point-line incidence graph of $\mathbb{F}_{q}^{d}$ by leveraging a new connection with the Furstenberg-S\'ark\"ozy problem from…

Let $V$ denote an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. For an $m$-dimensional $\mathbb{F}_q$-subspace $U$ of $V$ assume that $\dim_q \left(\langle {\bf v}\rangle_{\mathbb{F}_{q^n}} \cap U\right) \geq 2$ for each non zero vector…

A subfield $K$ of $\bar{\mathbb{Q}}$ is $large$ if every smooth curve $C$ over $K$ with a rational point has infinitely many rational points. A subfield $K$ of $\bar{\mathbb{Q}}$ is $big$ if for every positive integer $n$, $K$ contains a…

Let F_q be a finite field with odd q elements. In this article, we prove that if E \subseteq \mathbb F_q^d, d\ge 2, and |E|\ge q, then there exists a set Y \subseteq \mathbb F_q^d with |Y|\sim q^d$ such that for all y\in Y, the number of…

For any two integers $k,n$, $2\leq k\leq n$, let $f:(\mathbb{C}^*)^n\rightarrow\mathbb{C}^k$ be a generic polynomial map with given Newton polytopes. It is known that points, whose fiber under $f$ has codimension one, form a finite set…

$\lambda$-quiddities of size $n$ are $n$-tuples of elements from a fixed set that are solutions to a matrix equation which is fundamental in the study of the combinatorics of the modular group and Coxeter's friezes. To gain further insight…

In this paper, we first determine the minimum possible size of an Fq-linear set of rank k in PG(1, q^n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a…

We consider the dimensions of finite type of representations of a partially ordered set, i.e. such that there is only finitely many isomorphism classes of representations of this dimension. We give a criterion for a dimension to be of…

Let F be a family of subsets of an n-element set not containing four distinct members such that A union B is contained in C intersect D. It is proved that the maximum size of F under this condition is equal to the sum of the two largest…

We prove that a subset of $\mathbb{F}_q^n$ that contains a hyperplane in any direction has size at least $q^{n}-O(q^2)$.