Sascha Kurz
We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to provide new exact values for the minimum diameter of integral point sets.
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We give an attempt to build a classification of planar integral point sets. For two obtained classes, we provide general constructions of upper bounds for minimal diameter of integral point sets in higher dimensions of certain cardinality. Employing machine search, we give the examples of planar integral point sets to build such bounds.
A planar integral point set is a set of non-collinear points in plane such that for any pair of the points the Euclidean distance between the points is integral. We discuss the classification of planar integral point sets and provide examples of sets that are not covered by the existent classification.
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 s…
There are many papers studying properties of point sets in the Euclidean space $\mathbb{E}^m$ or on integer grids $\mathbb{Z}^m$, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of $\mathbb{Z} / \mathbb{Z}n$, and study the properties of the resulting combinatorial structures.
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 s…
A point set $M$ in the Euclidean plane is called a planar integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on a straight line. A planar integral point set is called to be in semi-general position, if it does not contain collinear triples. The existing lower bound for mininum diameter of planar integral point sets is linear. We prove a new lower bound for mininum diameter of planar integral point sets in semi-general position that is better than linear.
Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter $d(2,n)$ of a plane integral point set consisting of $n$ points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter c…
There are two known families of maximum scattered $\mathbb{F}_q$-linear sets in $PG(1,q^t)$: the linear sets of pseudoregulus type and for $t\geq 4$ the scattered linear sets found by Lunardon and Polverino. For $t=4$ we show that these are the only maximum scattered $\mathbb{F}_q$-linear sets and we describe the orbits of these linear sets under the groups $PGL(2,q^4)$ and $PΓL(2,q^4)$.
We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean spaces $\mathbb{E}^m$. We determine their maximal cardinality $\mathcal{I}(\mathbb{F}_q,2)$. For arbitrary commutative rings $\mathcal{R}$ instead of $\mathbb{F}_q$ or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.
A spherical three-distance set is a finite collection $X$ of unit vectors in $\mathbb{R}^{n}$ such that for each pair of distinct vectors has three inner product values. We use the semidefinite programming method to improve the upper bounds of spherical three-distance sets for several dimensions. We obtain better bounds in $\mathbb{R}^7$, $\mathbb{R}^{20}$, $\mathbb{R}^{21}$, $\mathbb{R}^{23}$, $\mathbb{R}^{24}$ and $\mathbb{R}^{25}$. In particular, we prove that maximum size of spherical three-distance sets is $2300$ in $\mathbb R^{23}$.
We prove that a subset of $\mathbb{F}_q^n$ that contains a hyperplane in any direction has size at least $q^{n}-O(q^2)$.
A mixed graph is said to be \textit{integral} if all the eigenvalues of its Hermitian adjacency matrix are integer. The \textit{mixed circulant graph} $Circ(\mathbb{Z}_n,\mathcal{C})$ is a mixed graph on the vertex set $\mathbb{Z}_n$ and edge set $\{ (a,b): b-a\in \mathcal{C} \}$, where $0\not\in \mathcal{C}$. If $\mathcal{C}$ is closed under inverse, then $Circ(\mathbb{Z}_n,\mathcal{C})$ is called a \textit{circulant graph}. We express the eigenvalues of $Circ(\mathbb{Z}_n,\mathcal{C})$ in terms of primitive $n$-th roots of unity, and find a sufficient condition for integrality of the eigenva…
Ehrhart polynomials are extensively-studied structures that interpolate the discrete volume of the dilations of integral $n$-polytopes. The coefficients of Ehrhart polynomials, however, are still not fully understood, and it is not known when two polytopes have equivalent Ehrhart polynomials. In this paper, we establish a relationship between Ehrhart-equivalence and other forms of equivalence: the $\operatorname{GL}_n(\mathbb{Z})$-equidecomposability and unimodular equivalence of two integral $n$-polytopes in $\mathbb{R}^n$. We conjecture that any two Ehrhart-equivalent integral $n$-polytopes …
For each natural number $d$, we introduce the concept of a $d$-cap in $\mathbb{F}_3^n$. A subset of $\mathbb{F}_3^n$ is called a $d$-cap if, for each $k = 1, 2, \dots, d$, no $k+2$ of the points lie on a $k$-dimensional flat. This generalizes the notion of a cap in $\mathbb{F}_3^n$. We prove that the $2$-caps in $\mathbb{F}_3^n$ are exactly the Sidon sets in $\mathbb{F}_3^n$ and study the problem of determining the size of the largest $2$-cap in $\mathbb{F}_3^n$.
A $\textit{ladder}$ is a set $S \subseteq \mathbb Z^+$ such that any finite coloring of $\mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $\mathbb Z^+$ itself is a ladder. We also discuss variants of ladders, namely $\textit{accessible}$ and $\textit{walkable}$ sets, which are sets $S$ such that any coloring of $\mathbb Z$ contains arbitrarily long (for accessible sets) or infinite (for walkable sets) monochromatic sequences with consecutive differences in $S$. We show that sets with upper density 1 a…
Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\{p_1,...,p_n\}\subset\mathbb{Z}^2$ a maximal integral point set over $\mathbb{Z}^2$ if all pairwise distances are integral and every additional point $p_{n+1}$ destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove …
We study generating series of Gromov-Witten invariants of $E\times\mathbb{P}^1$ and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each summand corresponds to a certain type of graph which we call a pearl chain. The individual summands are --- just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals --- complex analytic path integrals involving a product of propagators (equal to the Weierstrass-$\wp$-…
We show that sets avoiding 6-term arithmetic progressions in $\mathbb{Z}_6^n$ have size at most $5.709^n$. It is also pointed out that the "product construction" does not work in this setting, specially, we show that for the extremal sizes in small dimensions we have $r_6(\mathbb{Z}_6)=5$, $r_6(\mathbb{Z}_6^2)=25$ and $ 116\leq r_6(\mathbb{Z}_6^3)\leq 124$.
On Clusters and Exceptional Sets in Types $\mathbb{A}$ and $\tilde{\mathbb{A}}$
该文研究$\u2115$型与$\tilde{\u2115}$型的丛和例外集，证明$\tilde{\u2115}$型丛的无限族数由卡特兰数计数，明确相对投射性良定义，完成两类箭图例外集的相关计数。
Additively stable sets, critical sets for the 3k-4 theorem in $\mathbb{Z}$ and $\mathbb{R}$
本文研究整数与实数中的加法左稳定集，给出其在初始区间内密度的精确上界及达到该界的构造，并用于揭示 Freiman $3k-4$ 定理临界集的结构。
Motzkin posed the problem of finding the maximal density $μ(M)$ of sets of integers in which the differences given by a set $M$ do not occur. The problem is already settled when $|M|\leq 2$ and $M$ is a finite arithmetic progression. In this paper, we determine $μ(M)$ when $M$ has some other structure. For example, we determine $μ(M)$ when $M$ is a finite geometric progression.
On the characteristic and diameter of planar integral point sets
本文改进了半一般位置整数点集在特征为4k+1或4k+2时直径的下界，通过结合双曲线和网格工具集实现。
Lattice-free sets (convex subsets of $\mathbb{R}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral lattice-free polyhedra which are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra $\mathbb{Z}^d$-maximal. It is known that, for fixed $d$, the family $\mathbb{Z}^d$-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cutt…
Strong marker sets for arbitrary generating sets of $\mathbb Z^n$
本文通过幺模坐标变换，将任意有限生成集转化为全支撑位置，从而将高-王定理从特殊支撑情形推广到一般有限生成集。
We prove a general theorem for constructing integral quantum cluster algebras over ${\mathbb{Z}}[q^{\pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over ${\mathbb{Z}}[q^{\pm 1/2}]$. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that for every symmetrizable Kac-Moody algebra ${\mathfrak{g}}$ and Weyl group element $w$, the…
Let $K$ and $L$ be origin-symmetric convex integer polytopes in $\mathbb{R}^n$. We study a discrete analogue of the Aleksandrov projection problem. If for every $u\in \mathbb{Z}^n$, the sets $(K\cap \mathbb{Z}^n)|u^\perp$ and $(L\cap \mathbb{Z}^n)|u^\perp$ have the same number of points, is then $K=L$? We give a positive answer to this problem in $\mathbb{Z}^2$ under an additional hypothesis that $(2K\cap \mathbb{Z}^2)|u^\perp$ and $(2L\cap \mathbb{Z}^2)|u^\perp$ have the same number of points.
We present a sharp extension of a result of Bourgain on finding configurations of $k+1$ points in general position in measurable subset of $\mathbb{R}^d$ of positive upper density whenever $d\geq k+1$ to all proper $k$-degenerate distance graphs.
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