In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in \cite{HI07} and \cite{HIKR07}. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions.
In particular, our machinery enables us to prove that if $E \subset {\Bbb F}_q^d$, $d \ge 4$, the $d$-dimensional vector space over a finite field ${\Bbb F}_q$, of size much greater than $q^{\frac{d}{2}}$, and if $E$ is a product set, then the set of volumes of $d$-dimensional parallelepipeds determined by $E$ covers ${\Bbb F}_q$. This result is sharp as can be seen by taking $E$ to equal to $A \times A \times ... \times A$, where $A$ is a sub-field of ${\Bbb F}_q$ of size $\sqrt{q}$. In three dimensions we establish the same result if $|E| \gtrsim q^{15/8}$. We prove in three dimensions that the set of volumes covers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^{3/2}$. Finally we show that in three dimensions the set of volumes covers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^2$, without any further assumptions on $E$, which is again sharp as taking $E$ to be a 2-plane through the origin shows.
For any finite poset P we introduce a homogeneous space as a quotient of the general linear group with the incidence group of P. When P is a chain this quotient is a flag variety; for the trivial poset our construction gives a variety recently introduced in [20]. Moreover we provide decompositions for any set in a projective space, induced by the action of the incidence group of a suitable poset. In the classical cases of Grassmannians and flag varieties we recover, depending on the choice of the poset, the partition into Schubert cells and the matroid strata. Our general framework produces, f…
The incidence energy $\mathscr{IE}(G)$ of a graph $G$, defined as the sum of the singular values of the incidence matrix of a graph $G$, is a much studied quantity with well known applications in chemical physics. The Laplacian-energy-like invariant of $G$ is defined as the sum of square roots of the Laplacian eigenvalues. In this paper, we obtain the closed-form formulae expressing the incidence energy and the Laplacian-energy-like invariant of the Union Jack lattice. Moreover, the explicit asymptotic values of these quantities are calculated by utilizing the applications of analysis approach…
We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdo…
The point-plane incidence theorem states that the number of incidences between $n$ points and $m\geq n$ planes in the projective three-space over a field $F$, is $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear points, with the extra condition $n< p^2$ if $F$ has characteristic $p>0$. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in $F^2$, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in $…
We improve the range of exponents for the restriction problem for the 3-d paraboloid over finite fields. The key new ingredient is a variant of the Bourgain-Katz-Tao finite field incidence theorem derived from sum-product estimates. In prime order fields, we give an explicit relationship between the exponent in this incidence theorem and restriction estimates for the paraboloid.
Let $q$ be a power of a prime and $\mathbb{F}_q$ the finite field consisting of $q$ elements. We prove explicit upper bounds on the number of incidences between lines and Cartesian products in $\mathbb{F}_q^2$. We also use our results on point-line incidences to give new sum-product type estimates concerning sums of reciprocals.
Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$ and edge set $E(G) = \{e_1, e_2,\ldots, e_m\}$. Similar to the Randić matrix, here we introduce the Randić incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $n\times m$ matrix whose $(i, j)$-entry is $(d_i)^{-\frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise. Naturally, the Randić incidence energy $I_RE$ of $G$ is the sum of the singular values of $I_R(G)$. We establish lower and upper bounds for the Randić incidence energy. Graphs for which these bounds are best possible are cha…
In this paper we establish an improved bound for the number of incidences between a set $P$ of $m$ points and a set $H$ of $n$ planes in $\mathbb R^3$, provided that the points lie on a two-dimensional nonlinear irreducible algebraic variety $V$ of constant degree. Specifically, the bound is $$ O\left( m^{2/3}n^{2/3} + m^{6/11}n^{9/11}\log^β(m^3/n) + m + n + \sum_\ell |P_\ell|\cdot |H_\ell| \right) , $$ where the constant of proportionality and the constant exponent $β$ depend on the degree of $V$, and where the sum ranges over all lines $\ell$ that are fully contained in $V$ and contain at le…
Let $X$ be a finite connected poset and $K$ a field. We study the question, when all Lie automorphisms of the incidence algebra $I(X,K)$ are proper. Without any restriction on the length of $X$ we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of $X$. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns and ordinal sums of two antichains we give a complete answer.
We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most $O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive characteristic $p$, we assume $m^{-2}n^{13}\ll p^{15}$). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduc…
In this paper we show that the number of distinct distances determined by a set of $n$ points on a constant-degree two-dimensional algebraic variety $V$ (i.e., a surface) in $\mathbb R^3$ is at least $Ω\left(n^{7/9}/{\rm polylog} \,n\right)$. This bound is significantly larger than the conjectured bound $Ω(n^{2/3})$ for general point sets in $\mathbb R^3$. We also show that the number of unit distances determined by $n$ points on a surface $V$, as above, is $O(n^{4/3})$, a bound that matches the best known planar bound, and is worst-case tight in 3-space. This is in sharp contrast with the bes…
We show that any finite set S in a characteristic zero integral domain can be mapped to the finite field of order p, for infinitely many primes p, preserving all algebraic incidences in S. This can be seen as a generalization of the well-known Freiman isomorphism lemma, and we give several combinatorial applications (such as sum-product estimates).
In this paper, we prove a new point-sphere incidence bound in vector spaces over finite fields. More precisely, let $P$ be a set of points and $S$ be a set of spheres in $\mathbb{F}_q^d$. Suppose that $|P|, |S|\le N$, we prove that the number of incidences between $P$ and $S$ satisfies \[I(P, S)\le N^2q^{-1}+q^{\frac{d-1}{2}}N,\] under some conditions on $d, q$, and radii. This improves the known upper bound $N^2q^{-1}+q^{\frac{d}{2}}N$ in the literature. As an application, we show that for $A\subset \mathbb{F}_q$ with $q^{1/2}\ll |A|\ll q^{\frac{d^2+1}{2d^2}}$, one has \[\max \left\lbrace |A+…
In this paper we obtain a new sum-product estimate in prime fields. In particular, we show that if $A\subseteq \mathbb{F}_p$ satisfies $|A|\le p^{64/117}$ then $$ \max\{|A\pm A|, |AA|\} \gtrsim |A|^{39/32}. $$ Our argument builds on and improves some recent results of Shakan and Shkredov which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^+(P)$ of some subset $P\subseteq A+A$. Our main novelty comes from reducing the estimation of $E^+(P)$ to a point-plane incidence bound of Rudnev rather than a point line incidence bound of Stevens and de …
We study the sum-product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum-product bounds that depend on these parameters. For the dual numbers we expose a range where the minimum value of $\max\{|A+A|,|AA|\}$ is neither close to $|A|$ nor to $|A|^2$. To obtain our main sum-product bound, we extend Elekes' sum-product technique that relies on point-line…
This thesis establishes new quantitative records in several problems of incidence geometry and growth. After the necessary background in Chapters 1, 2 and 3, the following results are proven. Chapter 4 gives new results in the incidence geometry of a plane determined by a finite field of prime order. These comprise a new upper bound on the total number of incidences determined by finitely many points and lines, and a new estimate for the number of distinct lines determined by a finite set of non-collinear points. Chapter 5 gives new results on expander functions. First, a new bound is establis…
In this paper we study incidences for hyperbolas in $\mathbf{F}_p$ and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of Kloosterman sums.
The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of Rudnev and Shkredov. We bound the energy of several systems of lines, in some cases obtaining a better energy bound than the corresponding bounds obtained by Rudnev and Shkredov by exploiting a connection with collinear quadruples. Our motivation for seeking to generalise and improve the incidence bound obtained by Rudnev and Shkredov comes from possible applic…
This is an expository survey on recent sum-product results in finite fields. We present a number of sum-product or "expander" results that say that if $|A| > p^{2/3}$ then some set determined by sums and product of elements of $A$ is nearly as large as possible, and if $|A|<p^{2/3}$ then the set in question is significantly larger that $A$. These results are based on a point-plane incidence bound of Rudnev, and are quantitatively stronger than a wave of earlier results following Bourgain, Katz, and Tao's breakthrough sum-product result. In addition, we present two geometric results: an inciden…
Let $R$ be a finite valuation ring of order $q^r.$ Using a point-plane incidence estimate in $R^3$, we obtain sum-product type estimates for subsets of $R$. In particular, we prove that for $A\subset R$, $$|AA+A|\gg \min\left\{q^{r}, \frac{|A|^3}{q^{2r-1}}\right\}.$$ We also show that if $|A+A||A|^{2}>q^{3r-1}$, then $$|A^2+A^2||A+A|\gg q^{\frac{r}{2}}|A|^{\frac{3}{2}}.$$
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…
There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized Möbius function. Under the product this generalized Möbius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the Möbius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical Möbius functions. Using this generalized Möbi…
Incidence theorems for multivariate polynomials over finite fields
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究院）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education（教育部非线性期望前沿科学中心）；Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）
The aim of the dissertation is threefold: the first two parts are devoted to explore the Fourth Moment Theorem and universality properties for homogeneous sums, while the last part approaches the classical theory of orthogonal polynomials via the invariant theory of binary forms. Specifically, Part I deals with the Lindberg method of influence functions in free probability spaces. By providing a general multidimensional invariance principle for homogeneous sums in freely independent variables, a class of universal laws for semicircular and free Poisson approximations will be derived that satis…
This paper aims to study in more depth the relation between growth in matrix groups ${\rm SL_2}(\mathbf{F})$ and ${\rm Aff}(\mathbf{F})$ over a field $\mathbf{F}$ by multiplication and geometric incidence estimates, associated with the sum-product phenomenon over $\mathbf{F}$. It presents streamlined proofs of Helfgott's theorems on growth in the $\mathbf{F}_p$-case, which avoid sum-product estimates. For ${\rm SL_2}(\mathbf{F}_p)$, for sets exceeding in size some absolute constant, we improve the lower bound $\frac{1}{1512}$ for the growth exponent, due to Kowalski, to $\frac{1}{21}.$ For the…
A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D(X) and D(Y) are equivalent as triangulated categories. We give explicit combinatorial properties of a poset which are invariant under derived equivalence, among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. Then we construct, for any closed subset Y of X, a strongly exceptional collection in D(X)…
We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known, they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. Our concern is with discrepancies of distributions in metric balls and sums of pairwise distances between points of distributions in such spaces. Using the geometric features of two-point spaces, we show that Stolarsky's invariance principle, well-known for the Euclidean spheres, can be extended to all projecti…
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）
We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_p$. They are unified by the theme of "breaking the $3/2$ threshold", epitomising the previous state of the art. These estimates stem from specially suited applications of incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. We establish the estimate $|R[A]| \gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios defined by triples of elements of a (sufficiently small if $\mathbb{F}$ has…
