ID: 2310.07964

Exploration on Incidence Geometry and Sum-Product Phenomena

October 12, 2023

View on ArXiv
Sung-Yi Liao
Mathematics
Combinatorics
Number Theory

In additive combinatorics, Erd\"{o}s-Szemer\'{e}di Conjecture is an important conjecture. It can be applied to many fields, such as number theory, harmonic analysis, incidence geometry, and so on. Additionally, its statement is quite easy to understand, while it is still an open problem. In this dissertation, we investigate the Erd\"{o}s-Szemer\'{e}di Conjecture and its relationship with several well-known results in incidence geometry, such as the Szemer\'{e}di-Trotter Incidence Theorem. We first study these problems in the setting of real numbers and focus on the proofs by Elekes and Solymosi on sum-product estimates. After introducing these theorems, our main focus is the Erd\"{o}s-Szemer\'{e}di Conjecture in the setting of $\mathbb{F}_p$. We aim to adapt several ingenious techniques developed for real numbers to the case of finite fields. Finally, we obtain a result in estimating the number of bisectors over the ring $\mathbb{Z}/p^3\mathbb{Z}$ with $p$ a $4n+3$ prime.

Similar papers 1

Multi-parameter Szemer\'{e}di-Trotter-type theorems and applications in finite fields

July 11, 2023

88% Match
Hung Le, Steven Senger, Minh-Quan Vo
Combinatorics

We prove some novel multi-parameter point-line incidence estimates in vector spaces over finite fields. While these could be seen as special cases of higher-dimensional incidence results, they outperform their more general counterparts in those contexts. We go on to present a number of applications to illustrate their use in combinatorial problems from geometry and number theory.

Find SimilarView on arXiv

Incidence Theorems and Their Applications

August 24, 2012

87% Match
Zeev Dvir
Combinatorics
Computational Complexity

We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: (1) Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc..), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will ...

Find SimilarView on arXiv

Szemer\'{e}di-Trotter type results in arbitrary finite fields

August 16, 2018

87% Match
Ali Mohammadi
Combinatorics

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.

Find SimilarView on arXiv

Generalized incidence theorems, homogeneous forms, and sum-product estimates in finite fields

January 4, 2008

87% Match
David Covert, Derrick Hart, Alex Iosevich, ... , Rudnev Misha
Combinatorics
Classical Analysis and ODEs

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 met...

Find SimilarView on arXiv

New quantitative estimates on the incidence geometry and growth of finite sets

January 21, 2013

87% Match
Timothy G. F. Jones
Combinatorics

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 ...

Find SimilarView on arXiv
Ilya D. Shkredov
Number Theory
Combinatorics
Group Theory

In our paper, we introduce a new method for estimating incidences via representation theory. We obtain several applications to various sums with multiplicative characters and to Zaremba's conjecture from number theory.

Point-plane incidences and some applications in positive characteristic

June 9, 2018

86% Match
Misha Rudnev
Combinatorics
Classical Analysis and ODEs

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 Szemer\'edi-Trotter type bound for point-line incidences in $F^2$, due to Stevens and de Zeeuw. This review focuses...

Find SimilarView on arXiv

A new sum-product estimate in prime fields

July 29, 2018

86% Match
Changhao Chen, Bryce Kerr, Ali Mohammadi
Combinatorics
Number Theory

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 estim...

Find SimilarView on arXiv

Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture

July 24, 2007

85% Match
Derrick Hart, Alex Iosevich, ... , Rudnev Misha
Classical Analysis and ODEs
Combinatorics

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...

Find SimilarView on arXiv

On a paucity result in Incidence Geometry

January 20, 2022

85% Match
Ilya D. Shkredov
Number Theory
Combinatorics

We obtain some asymptotic formulae (with power savings in their error terms) for the number of quadruples in the Cartesian product of an arbitrary set $A \subset \mathbf{R}$ and for the number of quintuplets in $A\times A$ for any subset $A$ of the prime field $\mathbf{F}_p$. Also, we obtain some applications of our results to incidence problems in $\mathbf{F}_p$.

Find SimilarView on arXiv