TAILIEUCHUNG - Đề tài "The Erd˝os-Szemer´edi problem on sum set and product set"

The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd˝s-Szemer´di [E-S]. (see also [El], [T], o e and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A + A small and then deriving that the product set AA is large (using Freiman’s structure theorem) (cf. [N-T], [Na3]). We follow the. | Annals of Mathematics The Erd os-Szemer edi problem on sum set and product set By Mei-Chu Chang Annals of Mathematics 157 2003 939 957 The Erdos-Szemeredi problem on sum set and product set By Mei-Chu Chang Summary The basic theme of this paper is the fact that if A is a finite set of integers then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erdos-Szemeredi E-S . see also El T and K-T for related aspects. Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A A small and then deriving that the product set AA is large using Freiman s structure theorem cf. N-T Na3 . We follow the reverse route and prove that if IAA c A then A A c A 2 see Theorem 1 . A quantitative version of this phenomenon combined with the Pliinnecke type of inequality due to Ruzsa permit us to settle completely a related conjecture in E-S on the growth in k. If g k min A 1 A 1 over all sets A c z of cardinality A k and where A 1 respectively A 1 refers to the simple sum resp. product of elements of A. See . It was conjectured in E-S that g k grows faster than any power of k for k TO. Wp will Ỉ1ĨY1VÍ hnm thnt In ritỉA r -j Ợnt . mí1 whlr h is I Im mnin V V e w Hl pro v e neie biia b in g my Inln k see J- neoreiii J w inc 11 is b lie maul result of this paper. Introduction Let A B be finite sets of an abelian group. The sum set of A B is A B a b a G A b G B . We denote by hA A A h fold the h-fold sum of A. Partially supported by NSA. 940 MEI-CHU CHANG Similarly we can define the product set of A B and ft-fold product of A. AB ab I a G A b G B Ah A A h fold . If B b a singleton we denote AB by b A. In 1983 Erdos and Szemeredi E-S conjectured that for subsets of integers the sum set and the product set cannot both be small. Precisely they made the following conjecture. Conjecture 1 Erdos-Szemeredi . For any e 0 and any h G N there is

• Thạc sĩ - Tiến sĩ - Cao học
• product set
• subjects of mathematics
• Annals of Mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Product Suppor
• AT Command Syntax
• Generic Commands
• Control Commands
• 3GPP TS 27
• 007
• Network Service Commands
• scientific reports
• thesis report
• medical report
• The synthesis of the modified tetrapyrrole known asd1haem requires sev eral dedicated proteins which are coded for by a set of genes that are often found adjacent to the structural gene
• nirS
• for cytochrome cd1 nitrite reductase
• NirE
• the product of the first gene in the nirbiogenesis operon
• was anticipated to catalyse the conversion of uroporphyrinogen III into precorrin 2
• Lecture Linear algebra
• The Orthogonality and Least Squares
• The Scalar Product in R
• Orthogonal subspaces
• Orthonormal set
• The gram schmidt orthogonalization process
• Bài viết nghiên cứu khoa học
• Graded extension
• Monoidal category
• Factor set
• Crossed product
• Group cohomology
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center