Additive Number Theory

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Additive Number Theory The Classical Bases

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Author : Melvyn B. Nathanson
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14

Additive Number Theory The Classical Bases written by Melvyn B. Nathanson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.

[Hilbert's] style has not the terseness of many of our modem authors in mathematics, which is based on the assumption that printer's labor and paper are costly but the reader's effort and time are not. H. Weyl [143] The purpose of this book is to describe the classical problems in additive number theory and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools used to attack these problems. This book is intended for students who want to lel?Ill additive number theory, not for experts who already know it. For this reason, proofs include many "unnecessary" and "obvious" steps; this is by design. The archetypical theorem in additive number theory is due to Lagrange: Every nonnegative integer is the sum of four squares. In general, the set A of nonnegative integers is called an additive basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of A. Lagrange 's theorem is the statement that the squares are a basis of order four. The set A is called a basis offinite order if A is a basis of order h for some positive integer h. Additive number theory is in large part the study of bases of finite order. The classical bases are the squares, cubes, and higher powers; the polygonal numbers; and the prime numbers. The classical questions associated with these bases are Waring's problem and the Goldbach conjecture.

Additive Number Theory Of Polynomials Over A Finite Field

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Author : Gove W. Effinger
language : en
Publisher:
Release Date : 1991

Additive Number Theory Of Polynomials Over A Finite Field written by Gove W. Effinger and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991 with Mathematics categories.

This volume develops all the tools needed to provide modern, `adelic' proofs of the polynomial analogs to two of the most famous theorems of classical additive number theory: Vinogradov's 3-Primes Theorem and the Waring Problem.

Additive Number Theory

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Author : Melvyn Bernard Nathanson
language : en
Publisher:
Release Date : 1996

Additive Number Theory written by Melvyn Bernard Nathanson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with categories.

Additive Theory Of Prime Numbers

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Author : Luogeng Hua
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-12-04

Additive Theory Of Prime Numbers written by Luogeng Hua and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-12-04 with Mathematics categories.

Loo-Keng Hua was a master mathematician, best known for his work using analytic methods in number theory. In particular, Hua is remembered for his contributions to Waring's Problem and his estimates of trigonometric sums. Additive Theory of Prime Numbers is an exposition of the classic methods as well as Hua's own techniques, many of which have now also become classic. An essential starting point is Vinogradov's mean-value theorem for trigonometric sums, which Hua usefully rephrases and improves. Hua states a generalized version of the Waring-Goldbach problem and gives asymptotic formulas for the number of solutions in Waring's Problem when the monomial $x^k$ is replaced by an arbitrary polynomial of degree $k$. The book is an excellent entry point for readers interested in additive number theory. It will also be of value to those interested in the development of the now classic methods of the subject.

Combinatorial And Additive Number Theory Iii

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Author : Melvyn B. Nathanson
language : en
Publisher: Springer Nature
Release Date : 2019-12-10

Combinatorial And Additive Number Theory Iii written by Melvyn B. Nathanson and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-12-10 with Mathematics categories.

Based on talks from the 2017 and 2018 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 17 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003, the workshop series surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, commutative algebra and discrete geometry, and applications of logic and nonstandard analysis to number theory. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Additive Combinatorics

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Author : Terence Tao
language : en
Publisher: Cambridge University Press
Release Date : 2006-09-14

Additive Combinatorics written by Terence Tao and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-09-14 with Mathematics categories.

Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.

Structural Additive Theory

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Author : David J. Grynkiewicz
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-05-30

Structural Additive Theory written by David J. Grynkiewicz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-05-30 with Mathematics categories.

​Nestled between number theory, combinatorics, algebra and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e., sumsets and subsequence sums. This text is a hybrid of a research monograph and an introductory graduate textbook. With few exceptions, all results presented are self-contained, written in great detail, and only reliant upon material covered in an advanced undergraduate curriculum supplemented with some additional Algebra, rendering this book usable as an entry-level text. However, it will perhaps be of even more interest to researchers already in the field. The majority of material is not found in book form and includes many new results as well. Even classical results, when included, are given in greater generality or using new proof variations. The text has a particular focus on results of a more exact and precise nature, results with strong hypotheses and yet stronger conclusions, and on fundamental aspects of the theory. Also included are intricate results often neglected in other texts owing to their complexity. Highlights include an extensive treatment of Freiman Homomorphisms and the Universal Ambient Group of sumsets A+B, an entire chapter devoted to Hamidoune’s Isoperimetric Method, a novel generalization allowing infinite summands in finite sumset questions, weighted zero-sum problems treated in the general context of viewing homomorphisms as weights, and simplified proofs of the Kemperman Structure Theorem and the Partition Theorem for setpartitions.

Combinatorial And Additive Number Theory

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Author : Melvyn B. Nathanson
language : en
Publisher: Springer
Release Date : 2014-10-18

Combinatorial And Additive Number Theory written by Melvyn B. Nathanson and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-18 with Mathematics categories.

This proceedings volume is based on papers presented at the Workshops on Combinatorial and Additive Number Theory (CANT), which were held at the Graduate Center of the City University of New York in 2011 and 2012. The goal of the workshops is to survey recent progress in combinatorial number theory and related parts of mathematics. The workshop attracts researchers and students who discuss the state-of-the-art, open problems and future challenges in number theory.

Combinatorial Number Theory And Additive Group Theory

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Author : Alfred Geroldinger
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-04-15

Combinatorial Number Theory And Additive Group Theory written by Alfred Geroldinger and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-04-15 with Mathematics categories.

Additive combinatorics is a relatively recent term coined to comprehend the developments of the more classical additive number theory, mainly focussed on problems related to the addition of integers. Some classical problems like the Waring problem on the sum of k-th powers or the Goldbach conjecture are genuine examples of the original questions addressed in the area. One of the features of contemporary additive combinatorics is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, convex geometry, graph theory, probability theory, algebraic geometry or ergodic theory. This book gathers the contributions of many of the leading researchers in the area and is divided into three parts. The two first parts correspond to the material of the main courses delivered, Additive combinatorics and non-unique factorizations, by Alfred Geroldinger, and Sumsets and structure, by Imre Z. Ruzsa. The third part collects the notes of most of the seminars which accompanied the main courses, and which cover a reasonably large part of the methods, techniques and problems of contemporary additive combinatorics.

Elementary Methods In Number Theory

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Author : Melvyn B. Nathanson
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-01-11

Elementary Methods In Number Theory written by Melvyn B. Nathanson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-01-11 with Mathematics categories.

This basic introduction to number theory is ideal for those with no previous knowledge of the subject. The main topics of divisibility, congruences, and the distribution of prime numbers are covered. Of particular interest is the inclusion of a proof for one of the most famous results in mathematics, the prime number theorem. With many examples and exercises, and only requiring knowledge of a little calculus and algebra, this book will suit individuals with imagination and interest in following a mathematical argument to its conclusion.