Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Plücker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is \emph{non-degenerate} with respect to $L$ (see below). We show:
\medskip \noindent{\bf (1)} If $T$ is two-dimensional, the number of $r$-rich points (points incident to at least $r$ lines of $L$) is $O(n^{4/3+ε}/r^2)$, for $r \ge 3$ and for any $ε>0$, and, if at most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at most $O(n^{4/3+ε})$ $2$-rich points. For $r$ larger than some sufficiently large constant, the number of $r$-rich points is also $O(n/r)$.
As an application, we deduce (with an $ε$-loss in the exponent) the bound obtained by Pach and de Zeeuw (2107) on the number of distinct distances determined by $n$ points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle.
\medskip \noindent{\bf (2)} If $T$ is two-dimensional, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O(m+n)$.
\medskip \noindent{\bf (3)} If $T$ is three-dimensional and nonlinear, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O\left(m^{3/5}n^{3/5} + (m^{11/15}n^{2/5} + m^{1/3}n^{2/3})s^{1/3} + m + n \right)$, provided that no plane contains more than $s$ of the points. When $s = O(\min\{n^{3/5}/m^{2/5}, m^{1/2}\})$, the bound becomes $O(m^{3/5}n^{3/5}+m+n)$.
As an application, we prove that the number of incidences between $m$ points and $n$ lines in $R^4$ contained in a quadratic hypersurface (which does not contain a hyperplane) is $O(m^{3/5}n^{3/5} + m + n)$.
The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.
The Pascal matrix, $P$, is an upper diagonal matrix whose entries are the binomial coefficients. In 1993 Call and Velleman demonstrated that it satisfies the beautiful relation $P=\exp(H)$ in which $H$ has the numbers 1, 2, 3, etc. on its superdiagonal and zeros elsewhere. We generalize this identity to the incidence algebras $I(A^*)$ and $I(\mathcal{S})$ of functions on words and permutations, respectively. In $I(A^*)$ the entries of $P$ and $H$ count subwords; in $I(\mathcal{S})$ they count permutation patterns. Inspired by vincular permutation patterns we define what it means for a subword …
We prove an incidence result counting the $k$-rich $δ$-tubes induced by a well-spaced set of $δ$-atoms. Our result coincides with the bound that would be heuristically predicted by the Szemerédi--Trotter Theorem and holds in all dimensions $d \geq 2$.
We prove three theorems giving extremal bounds on the incidence structures determined by subsets of the points and blocks of a balanced incomplete block design (BIBD). These results generalize and strengthen known bounds on the number of incidences between points and $m$-flats in affine geometries over finite fields. First, we show an upper bound on the number of incidences between sufficiently large subsets of the points and blocks of a BIBD. Second, we show that a sufficiently large subset of the points of a BIBD determines many t-rich blocks. Third, we show that a sufficiently large subset …
The classical 1991 result by Brightwell and Winkler states that the number of linear extensions of a poset is #P-complete. We extend this result to posets with certain restrictions. First, we prove that the number of linear extension for posets of height two is #P-complete. Furthermore, we prove that this holds for incidence posets of graphs. Finally, we prove that the number of linear extensions for posets of dimension two is #P-complete.
We propose a generalization of the classical point-line Cremona-Richmond configuration to a configuration of points and more dimensional subspaces of a projective space, and present them as geometric realizations of some interesting abstract incidence structures.
The main purpose of this paper is to provide threshold functions for the events that a random subset of the points of a finite vector space has certain properties related to point-flat incidences. Specifically, we consider the events that there is an $\ell$-rich $m$-flat with regard to a random set of points in $\mathbb{F}_q^n$, the event that a random set of points is an $m$-blocking set, and the event that there is an incidence between a random set of points and a random set of $m$-flats. One of our key ingredients is a stronger version of a recent result obtained by Chen and Greenhill (2021…
We prove new incidence bounds between a plane point set, which is a Cartesian product, and a set of translates $H$ of the hyperbola $xy=λ\neq 0$, over a field of asymptotically large positive characteristic $p$. They improve recent bounds by Shkredov, which are based on using explicit incidence estimates in the early terminated procedure of repeated applications of the Cauchy-Schwarz inequality, underlying many qualitative results related to growth and expansion in groups. The improvement -- both quantitative, plus we are able to deal with a general $H$, rather than a Cartesian product -- is m…
We prove that the number of incidences between $m$ points and $n$ bounded-degree curves with $k$ degrees of freedom in ${\mathbb R}^d$ is \[ I(m,n) =O\left(m^{\frac{k}{dk-d+1}+\varepsilon}n^{\frac{dk-d}{dk-d+1}}+ \sum_{j=2}^{d-1} m^{\frac{k}{jk-j+1}+\varepsilon}n^{\frac{d(j-1)(k-1)}{(d-1)(jk-j+1)}}q_j^{\frac{(d-j)(k-1)}{(d-1)(jk-j+1)}}+m+n\right), \] for any $\varepsilon>0$, where the constant of proportionality depends on $k, \varepsilon$ and $d$, provided that no $j$-dimensional surface of degree $\le c_j(k,d,\varepsilon)$, a constant parameter depending on $k$, $d$, $j$, and $\varepsilon$, …
We show that the maximum number of pairwise non-overlapping $k$-rich lenses (lenses formed by at least $k$ circles) in an arrangement of $n$ circles in the plane is $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{5/2}} + \frac{n}{k} \right)$, and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles that form it) is $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{3/2}} + n\right)$. Two independent proofs of these bounds are given, each interesting in its own right (so we believe). We then show that these bounds lead to the known bound of Agarwal et al. (JACM …
We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains Ω(n^{7/8}) points of P and no circle contains Ω(n^{5/6}) points of P. We rely on the bipartite and partial variant of the Elekes-Sharir framework that was presented by Sharir, Sheffer, and Solymosi in \cite{SSS13}. For the case of lines we combine this framework with a theorem from additive combinatorics, and for the case of circles we combine it with some basic algebraic geometry and a recen…
Packing sets under finite groups via algebraic incidence structures
Eötvös University（厄特沃什大学）；Alfréd Rényi Institute（阿尔弗雷德·雷尼研究所）；Hanoi University of Science and Technology（河内科技大学）；Department of Mathematics, University of Rochester（罗切斯特大学数学系）；Institute for Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）
Lonesum matrices are matrices that are uniquely reconstructible from their row and column sum vectors. These matrices are enumerated by the poly-Bernoulli numbers that are related to the multiple zeta values and have a rich literature in number theory. Combinatorially, lonesum matrices are in bijection with many other combinatorial objects: several permutation classes, other matrix classes, acyclic orientations in graphs etc. Motivated of these facts, we study in this paper lonesum matrices with restriction on the number of columns and rows of the same type.
Given two points $p,q$ in the real plane, the signed area of the rectangle with the diagonal $[pq]$ equals the square of the Minkowski distance between the points $p,q$. We prove that $N>1$ points in the Minkowski plane $\R^{1,1}$ generate $Ω(\frac{N}{\log{N}})$ distinct distances, or all the distances are zero. The proof follows the lines of the Elekes/Sharir/Guth/Katz approach to the Erd\H os distance problem, analysing the 3D incidence problem, arising by considering the action of the Minkowski isometry group $ISO^*(1,1)$. The signature of the metric creates an obstacle to applying the Guth…
Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families
South China Normal University（华南师范大学）；Linköping University（林雪平大学）；Jiangsu Provincial Scientific Research Center of Applied Mathematics（江苏省应用数学省级科研中心）；The Hong Kong Polytechnic University（香港理工大学）
For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied sparse sets provide a rich setting for additive number theory, and in this paper we study various questions relating to the appearance of arithmetic progressions in these sets. In particular, for all $b$ we determine exactly the maximal length of an arithmetic progression that omits a base-$b$ digit.
We derive an asymptotic formula for $A(n,j,r)$ the number of integer partitions of $n$ into at most $j$ parts each part $\le r$. We assume $j$ and $r$ are near their mean values. We also investigate the second largest part, the number of parts $\ge 2$, etc. We show that the fraction of the partitions of an even integer $n$ that are graphical, ie. whose parts form the degree sequence of a simple graph, is $O(\ln^{-1/2} n)$. Probabilistic results are used in our discussion of graphical partitions. The George Szekeres circle method is essential for our asymptotic results on partitions. We determi…
