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Erdos Distinct Distances Problem and Extensions over Finite Spaces

Van Thang Pham
EPFL, 2017
Thèse EPFL

Résumé

In this thesis we study a number of problems in Discrete Combinatorial Geometry in finite spaces. The contents in this thesis are structured as follows: In Chapter 1 we will state the main results and the notations which will be used throughout the thesis. Chapter 2 is a version of the paper entitled "Sumsets of the distance sets in finite spaces", which has been submitted for publication, (2017). Chapter 3 is a version of the paper entitled "Three-variable expanding polynomials and higher-dimensional distinct distances", which has been submitted for publication, co-authored with L. A. Vinh and de Zeeuw. The author was one of the main investigators of this chapter. Chapter 4 is a postprint version of the paper entitled "Distinct distances on regular varieties over finite fields", Journal of Number Theory, 173(2017), 602-613, co-authored with D. D. Hieu. The author was one of the main investigators of this chapter. Chapter 5 is a postprint version of the paper entitled " Incidences between points and generalized spheres over finite fields and related problems", Forum Mathematicum, Volume 29, Issue 2 (Mar 2017), co-authored with N. D. Phuong and L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 6 is a version of the paper entitled "Distinct spreads in finite spaces", which has been submitted for publication, co-authored with B. Lund and L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 7 is a version of the paper entitled "Paths in pseudo-random graphs", which has been submitted for publication, co-authored with L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 8 is a version of the paper entitled "Conditional expanding bounds for two-variable functions over arbitrary fields", which has been submitted for publication, co-authored with Hossein Nassajian Mojarrad. The author was one of the main investigators of this chapter. Chapter 9 is a postprint version of the paper entitled "A Szemeredi-Trotter type theorem, sum-product estimates in finite quasifields, and related results", Journal of Combinatorial Theory Series A, 147(2017), 55-74, co-authored with Michael Tait, Craig Timmons, Le Anh Vinh. The author was one of the main investigators of this chapter. The content of this chapter also appears in Michael Tait's Phd thesis. In Chapter 10, we will mention some open problems on Erd\H{o}s distinct distances problem and generalizations.

Official source

https://infoscience.epfl.ch/record/228901?ln=fr

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Concepts associés

Chargement

Publications associées

Chargement

Van Thang Pham
EPFL, 2017
Thèse EPFL

Résumé

Official source

https://infoscience.epfl.ch/record/228901?ln=fr

À propos de ce résultat

Concepts associés (19)

Concepts associés

Chargement

Publications associées (7)

Publications associées

Chargement

Incidences between points and generalized spheres over finite fields and related problems

Van Thang Pham

Let F-q be a finite field of q elements, where q is a large odd prime power and Q = a(1)x(1)(c1) + ..... + a(d)x(d)(cd) is an element of F-q[x(1) ,...,x(d)], where 2

Walter De Gruyter Gmbh2017

Chargement

Algebraic and topological methods in combinatorics

Adrian Claudiu Valculescu

The present thesis deals with problems arising from discrete mathematics, whose proofs make use of tools from algebraic geometry and topology. The thesis is based on four papers that I have co-authored, three of which have been published in journals, and one has been submitted for publication (and also appeared as a preprint on the arxiv, and as an extendend abstract in a conference). Specifically, we deal with the following four problems: \begin{enumerate} \item We prove that if

M\in \mathbb{C}^{2\times2}

is an invertible matrix, and

B_M:\mathbb{C}^2\times\mathbb{C}^2\to\mathbb{C}

is a bilinear form

B_M(p,q)=p^TMq

, then any finite set

S

contained in an irreducible algebraic curve

C

of degree

d

\mathbb{C}^2

determines

\Omega_d(|S|^{4/3})

distinct values of

B_M

, unless

C

is a line, or is linearly equivalent to a curve defined by an equation of the form

x^k=y^l

, with

k,l\in\mathbb{Z}\backslash\\{0\\}

, and

\gcd(k,l)=1

. \item We show that if we are given

m

points and

n

lines in the plane, then the number of distinct distances between the points and the lines is

\Omega(m^{1/5}n^{3/5})

, as long as

m^{1/2}\le n\le m^2

. Also, we show that if we are given

m

points in the plane, not all collinear, then the number of distances between these points and the lines that they determine is

\Omega(m^{4/3})

. We also study three-dimensional versions of the distinct point-line distances problem. \item We prove the lower bound

\Omega(|S|^4)

on the number of ordinary conics determined by a finite point set

S

\mathbb{R}^2

, assuming that

S

is not contained in a conic, and at most

c|S|

points of

S

lie on the same line (for some $0

EPFL2017

Chargement

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer

n\ge 3

, we define

e(n)

to be the minimum positive integer such that any set of

e(n)

points in general position in the plane contains

n

points in convex position. In 1935, Erd\H{o}s and Szekeres proved that

e(n) \le {2n-4 \choose n-2}+1

and later in 1961, they obtained the lower bound

2^{n-2}+1 \le e(n)

, which they conjectured to be optimal. We prove that

e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2

. In a recent breakthrough, Suk proved that

e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}

. We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let

c(n)

(and

c'(n)

) denote the smallest positive integer

N

such that any family of

N

pairwise disjoint convex bodies in general position (resp.,

N

convex bodies in general position, any pair of which share at most two boundary points) has an

n

members in convex position. We show that

c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}

. Given a point set

P

in the plane, an ordinary circle for

P

is defined as a circle containing exactly three points of

P

. We prove that any set of

n

points in the plane, not all on a line or a circle, determines at least

\frac{1}{4}n^2-O(n)

ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large

n

, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve

C\subset \mathbb{C}^2

of degree

d

and two finite sets

A,B\subset \mathbb{C}

, we have

|C\cap (A\times B)|=O_d(|A|+|B|)

. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection

|X\cap (P\times Q)|

for a variety

X\subset \mathbb{C}^4

and finite sets

P,Q\subset \mathbb{C}^2

. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that

\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}

for any small set

A\subset \mathbb{F}_p

and quadratic non-degenerate polynomial

f(x, y)\in \mathbb{F}_p[x, y]

. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term

\max \{ |A+A|, |A \cdot A |\}

. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on

\max\{ |A+B|, |f(B,C)|\}

, for a specific type of function

f

. Finally, we prove that the number of distinct cubic distances generated by any small set

A\times A\subset \mathbb{F}_p^2

\Omega(|A|^{8/7})

, which improves a result of Yazici et al.

EPFL2017

Chargement

Algebraic and topological methods in combinatorics

Adrian Claudiu Valculescu

M\in \mathbb{C}^{2\times2}

is an invertible matrix, and

B_M:\mathbb{C}^2\times\mathbb{C}^2\to\mathbb{C}

is a bilinear form

B_M(p,q)=p^TMq

, then any finite set

S

contained in an irreducible algebraic curve

C

of degree

d

\mathbb{C}^2

determines

\Omega_d(|S|^{4/3})

distinct values of

B_M

, unless

C

is a line, or is linearly equivalent to a curve defined by an equation of the form

x^k=y^l

, with

k,l\in\mathbb{Z}\backslash\\{0\\}

, and

\gcd(k,l)=1

. \item We show that if we are given

m

points and

n

lines in the plane, then the number of distinct distances between the points and the lines is

\Omega(m^{1/5}n^{3/5})

, as long as

m^{1/2}\le n\le m^2

. Also, we show that if we are given

m

points in the plane, not all collinear, then the number of distances between these points and the lines that they determine is

\Omega(m^{4/3})

. We also study three-dimensional versions of the distinct point-line distances problem. \item We prove the lower bound

\Omega(|S|^4)

on the number of ordinary conics determined by a finite point set

S

\mathbb{R}^2

, assuming that

S

is not contained in a conic, and at most

c|S|

points of

S

lie on the same line (for some $0

EPFL2017

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer

n\ge 3

, we define

e(n)

to be the minimum positive integer such that any set of

e(n)

points in general position in the plane contains

n

points in convex position. In 1935, Erd\H{o}s and Szekeres proved that

e(n) \le {2n-4 \choose n-2}+1

and later in 1961, they obtained the lower bound

2^{n-2}+1 \le e(n)

, which they conjectured to be optimal. We prove that

e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2

. In a recent breakthrough, Suk proved that

e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}

c(n)

(and

c'(n)

) denote the smallest positive integer

N

such that any family of

N

pairwise disjoint convex bodies in general position (resp.,

N

convex bodies in general position, any pair of which share at most two boundary points) has an

n

members in convex position. We show that

c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}

. Given a point set

P

in the plane, an ordinary circle for

P

is defined as a circle containing exactly three points of

P

. We prove that any set of

n

points in the plane, not all on a line or a circle, determines at least

\frac{1}{4}n^2-O(n)

ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large

n

C\subset \mathbb{C}^2

of degree

d

and two finite sets

A,B\subset \mathbb{C}

, we have

|C\cap (A\times B)|=O_d(|A|+|B|)

. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection

|X\cap (P\times Q)|

for a variety

X\subset \mathbb{C}^4

and finite sets

P,Q\subset \mathbb{C}^2

\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}

for any small set

A\subset \mathbb{F}_p

and quadratic non-degenerate polynomial

f(x, y)\in \mathbb{F}_p[x, y]

. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term

\max \{ |A+A|, |A \cdot A |\}

. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on

\max\{ |A+B|, |f(B,C)|\}

, for a specific type of function

f

. Finally, we prove that the number of distinct cubic distances generated by any small set

A\times A\subset \mathbb{F}_p^2

\Omega(|A|^{8/7})

, which improves a result of Yazici et al.

EPFL2017