给定一个大的有限点集，$P\subset \mathbb R^2$，我们得到的上界的点，确定一个给定的点积对的三元组的数量。也就是说，对于任何一对正实数$(α, β)$，我们限制了集合$$\left\{(p,q,r)\in P \times P \times P : p \cdot q = α, p \cdot r = β\right\}.$$的大小
英文摘要：
Given a large finite point set, $P\subset \mathbb R^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(α, β)$, we bound the size of the set $$\left\{(p,q,r)\in P \times P \times P : p \cdot q = α, p \cdot r = β\right\}.$$
Since 2002, the best known upper bound on the Ramsey numbers R n (3) = R(3,. .. , 3) is R n (3) $\le$ n!(e -- 1/6) + 1 for all n $\ge$ 4. It is based on the current estimate R 4 (3) $\le$ 62. We show here how any closing-in on R 4 (3) yields an improved upper bound on R n (3) for all n $\ge$ 4. For instance, with our present adaptive bound, the conjectured value R 4 (3) = 51 implies R n (3) $\le$ n!(e -- 5/8) + 1 for all n $\ge$ 4.
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.
We study a variant of Erd\H os' unit distance problem, concerning dot products between successive pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of nonzero dot products $(α_1,\ldots,α_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(A_1,\dots, A_{k+1})\in E^{k+1}$ satisfying $A_j \cdot A_{j+1}=α_j$ for every $1\leq j \leq k$.
我们得到的三元组的数量上的界限，确定一个给定的一对点的产品出现在一个向量空间在一个有限域或一个模块的集合上的整数模幂的素数。更准确地说，给定$E\subset \mathbb F_q^d$或$\mathbb Z_q^d$，我们为单元$α$和$β$提供集合\[\left\{(u,v,w)\in E \times E \times E : u\cdot v = α, u \cdot w = β\right\}\]大小的界限。
Size of dot product sets determined by pairs of subsets of vector spaces over finite fields
本文研究了有限域上两个向量空间子集生成的点积集的基数。我们注意到，一个集合的点积问题的结果可以简单地推广到两个集合。设E和F是有限域F_q上的d维向量空间F_q^d的子集。作为一个新的结果，我们证明了：如果E和F是抛物面的子集，且|E| |F| \geq C q^d对某个大的C>1，则|（英、法）|\geq c q，其中0<c<1。特别地，我们发现了点积集的大小与通过给定集合E中的原点和非零点的线的数目之间的联系。作为这一观察的应用，我们得到了广义点积集问题的更尖锐的结果。离散的傅里叶分析和几何观察在证明我们的结果中起着至关重要的作用。
We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime. Specifically, given a large finite set of points $E$, and a sequence of elements of the base field (or ring) $(α_1,\ldots,α_k)$, we give conditions guaranteeing the expected number of $(k+1)$-tuples of distinct points $(x_1,\dots, x_{k+1})\in E^{k+1}$ satisfying $x_j \cdot x_{j+1}=α_j$ for every $1\leq j \leq k$.
设$\mathbb{F}$是一个域，$P \subseteq \mathbb{F}^d$是一个有限点集，$α,β\in \mathbb{F} \setminus \{0\}$。我们研究的数量\[|Π_{α, β}| = \{(p,q,r) \in P \times P \times P \mid p \cdot q = α, p \cdot r = β\}.\]我们观察到的问题之间的连接，把一个上限$|Π_{α,β}|$和一个很好的研究问题之间的发病率点和超平面，并使用这种连接证明新的和加强的上界$|Π_{α,β}|$在各种设置。
We consider a problem related to the Erdoős Unit Distance Conjecture: How often can a single dot product configuration or a multiple dot product configuration occur over a Galoi Ring? We also find a bound on an inverse vector matrix multiplication problem.
For $k\geq3$, a collection of $k$ sets is said to form a \emph{weak $Δ$-system} if the intersection of any two sets from the collection has the same size. Erdős and Szemerédi asked about the size of the largest family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ that does not contain a weak $Δ$-system. In this note we improve upon the best upper bound of the author and Sawin from arXiv:1606.09575 and show that \[ |\mathcal{F}|\leq\left(\frac{2}{3}Θ(C)+o(1)\right)^{n} \] where $Θ(C)$ is the capset capacity. In particular, this shows that \[ |\mathcal{F}|\leq(1.8367\dots+o(1))^{n}. \]
Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,B\subset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $A\cup B$ is a total dominating set. A vertex partition $Ψ=\{C_1,C_2,\dots,C_k\}$ is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every $i\in\{1,2,\dots,k\}$ there exists a distinct $j\in\{1,2,\dots,k\}$ such that $C_i$ and $C_j$ form a total coalition. The maximum cardinality of a total coalition partition of $G$ is the total coalition number of $G$ and denoted …
Let $d \geq 1$ be an integer. From a set of $d$-dimensional vectors, we obtain a $d$-\dpg\ by letting each vector $\va^u$ correspond to a vertex $u$ and by adding an edge between two vertices $u$ and $v$ if and only if their dot product $\va^{u} \cdot \va^{v} \geq t$, for some fixed, positive threshold~$t$. Dot product graphs can be used to model social networks. Recognizing a $d$-dot product graph is known to be \NP-hard for all fixed $d\geq 2$. To understand the position of $d$-dot product graphs in the landscape of graph classes, we consider the case $d=2$, and investigate how $2$-dot produ…
Let $G$ be an abelian group. A set $A \subset G$ is a \emph{$B_k^+$-set} if whenever $a_1 + \dots + a_k = b_1 + \dots + b_k$ with $a_i, b_j \in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a $B_k^+$-set but the converse is not true in general. Determining the largest size of a $B_k$-set in the interval $\{1, 2, \dots, N \} \subset \integers$ or in the cyclic group $\integers_N$ is a well studied problem. In this paper we investigate the corresponding problem for $B_k^+$-sets. We prove non-trivial upper bounds on the maximum size of a $B_k^+$-set con…
We study a family of variants of Erd\H os' unit distance problem, concerning distances and dot products between pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, we look for bounds on how many subsets of $k$ points satisfy a set of relationships between point pairs based on distances or dot products. We survey some of the recent work in the area and present several new, more general families of bounds.
In this paper we consider four basic multidimensional matrix operations (outer product, Kronecker product, contraction, and projection) and two derivative operations (dot and circle products). We start with the interrelations between these operations and deduce some of their algebraic properties. Next, we study their action on $k$-stochastic matrices. At last, we prove several relations on the permanents of products of multidimensional matrices. In particular, we obtain that the permanent of the dot product of nonnegative multidimensional matrices is not less than the product of their permanen…
For fixed graphs $F_1,\dotsc,F_r$, we prove an upper bound on the threshold function for the property that $G(n,p) \to (F_1,\dotsc,F_r)$. This establishes the $1$-statement of a conjecture of Kohayakawa and Kreuter.
We present a family of sharpness examples for Falconer-type single dot product results. In particular, for $d\geq 2,$ for any $s<\frac{d+1}{2},$ we construct a Borel probability measure $μ$ satisfying the energy estimate $I_s(μ)<\infty,$ yet the estimate \begin{equation} (μ\times μ)\{(x,y):1\leq x\cdot y \leq 1+ε\} \leq Cε\end{equation} does not hold with constants independent of $ε$. It is known (\cite{EIT11}) that such an estimate always holds with $C$ independent of $ε$ if $I_{\frac{d+1}{2}}(μ)<\infty$. Thus our estimate proves the sharpness of the dimensional threshold in this result and g…
Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of $\mathbb{R}^m$ with respect to the portion of the graph of $F$ defined by $a\leq \log |s| \leq T$ is at most $O(1/(T-a))$. We conclude that if $I \subseteq \mathbb{R}^m$ has positive upper density, then the difference set $I-I$ contains vectors of the form $F(s)$ f…
Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an improved bound for the number of dot products determined by $P$, proving that, \[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the proof of this bound is a new superquadratic expander involving products and shifts. We prove that, for any finite set $X \subset \mathbb R$, there exist $z,z' \in X$ such that \[ \left|\frac{(zX+1)^{(2)}(z'X+1)^{(2)}}{(zX+1)^{(2)}(z'X+1)}\right| \gtrsim |X|^{5/2}. \] This is derived from a more general result concerning growth of sets defined via convexity and sum …
We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then $$N\le \exp(O(k^{1-c}))\,.$$
Given a set $E \subset {\Bbb F}_q^3$, where ${\Bbb F}_q$ is the field with $q$ elements. Consider a set of "classifiers" ${\mathcal H}^3_t(E)=\{h_y: y \in E\}$, where $h_y(x)=1$ if $x \cdot y=t$, $x \in E$, and $0$ otherwise. We are going to prove that if $|E| \ge Cq^{\frac{11}{4}}$, with a sufficiently large constant $C>0$, then the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t(E)$ is equal to $3$. In particular, this means that for sufficiently large subsets of ${\Bbb F}_q^3$, the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t(E)$ is the same as the Vapnik-Chervonenkis dimension of …
