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Register a new userMaximal integral point sets in affine planes over finite fields
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Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $mathbb{F}_q^2$ over a finite field $mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $mathbb{F}_q$. It turns out that integral point sets over $mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over $mathbb{F}_q$ for $qle 47$. Furthermore, we give two series of maximal integral point sets and prove their maximality.
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A $(k,m)$-Furstenberg set $S subset mathbb{F}_q^n$ over a finite field is a set that has at least $m$ points in common with a $k$-flat in every direction. The question of determining the smallest size of such sets is a natural generalization of the finite field Kakeya problem. The only previously known bound for these sets is due to Ellenberg-Erman and requires sophisticated machinery from algebraic geometry. In this work we give new, completely elementary and simple, proofs which significantly improve the known bounds. Our main result relies on an equivalent formulation of the problem using the notion of min-entropy, which could be of independent interest.
We say that $M$ and $S$ form a textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, while $0$ has no such representation. The splitting is called {it nonsingular} if $gcd(|G|, a) = 1$ for any $ain M$. In this paper, we focus our study on nonsingular splittings of cyclic groups. We introduce a new notation --direct KM logarithm and we prove that if there is a prime $q$ such that $M$ splits $mathbb{Z}_q$, then there are infinitely many primes $p$ such that $M$ splits $mathbb{Z}_p$.
Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the smallest possible diameter $d(m,n)$ for given parameters $m$ and $n$. We give some new upper bounds for the minimum diameter $d(m,n)$ and some exact values.
In this paper, we study dot-product sets and $k$-simplices in vector spaces over finite rings. We show that if $E$ is sufficiently large then the dot-product set of $E$ covers the whole ring. In higher dimensional cases, if $E$ is sufficiently large then the set of simplices and the set of dot-product simplices determined by $E$, up to congurence, have positive densities.
Let $mathbb{F}_q$ be a finite field of order $q$, and $P$ be the paraboloid in $mathbb{F}_q^3$ defined by the equation $z=x^2+y^2$. A tuple $(a, b, c, d)in P^4$ is called a non-trivial energy tuple if $a+b=c+d$ and $a, b, c, d$ are distinct. For $Xsubset P$, let $mathcal{E}^+(X)$ be the number of non-trivial energy tuples in $X$. It was proved recently by Lewko (2020) that $mathcal{E}^+(X)ll |X|^{frac{99}{41}}$ for $|X|ll q^{frac{26}{21}}$. The main purposes of this paper are to prove lower bounds of $mathcal{E}^+(X)$ and to study related questions by using combinatorial arguments and a weak hypergraph regularity lemma developed recently by Lyall and Magyar (2020).
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