Combinatorial and Additive Number Theory

Simultaneous systems of representatives and combinatorial number theory

Elementary Methods in Number Theory begins with first course in number for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.

New relationships between questions in combinatorial number theory are shown. The Erdos-Turan conjecture is shown to be equivalent to the finite problem; the Erdos-Turan conjecture for N is shown to follow from that for Z; and the additive number theory of the integers and that of the finite bit strings under bitwise mod 2 addition are contrasted. The dual space technique from coding theory is applied to the integer question, leading to a lower bound on the number of collisions. A subsidiary question in coding theory is raised, which is a refinement of the perfect code theorem. Finally some computations for cyclic codes are given.

Publisher Summary This chapter discusses the problems and results on combinatorial number theory. It discusses number theoretic problems that are of combinatorial nature. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers along with the properties of objects made out of integers. Integers can be considered either in themselves or as solutions to equations. Questions in number theory are often understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. Real numbers can also be studied in relation to rational numbers.

A method of constructing codes capable of correcting an arbitrary number of asymmetrical errors is given. The method is related to a problem in the additive theory of numbers.

Journal of the London Mathematical SocietyVolume s1-25, Issue 2 p. 115-119 Some aspects of Hardy's mathematical work (4) the Additive Theory of Numbers A. E. Ingham, A. E. InghamSearch for more papers by this author A. E. Ingham, A. E. InghamSearch for more papers by this author First published: April 1950 https://doi.org/10.1112/jlms/s1-25.2.115AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes1-25, Issue2April 1950Pages 115-119 RelatedInformation

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 to attack these problems. This book is intended for students who want to learn additive number theory, not for experts who already know it. The prerequisites for this book are undergraduate courses in number theory and real analysis.

Exact order of subsets of asymptotic bases in additive number theory

Linear extension of the Erdős–Heilbronn conjecture

In these notes we try to demonstrate the utility of the theory of theta functions in combinatorial number theory and complex analysis. The main idea is to use identities among theta functions to deduce either useful number-theoretic information related to representations as sums of squares and triangular numbers, statements concerning congruences, or statements concerning partitions of sets of integers. In complex analysis the main utility is in the theory of compact Riemann surfaces, with which we do not deal. We do show how identities among theta functions yield proofs of Picard’s theorem and a conformal map of the rectangle onto the disk.

Starting with a simple formulation accessible to all mathematicians, this second edition is designed to provide a thorough introduction to nonstandard analysis. Nonstandard analysis is now a well-developed, powerful instrument for solving open problems in almost all disciplines of mathematics; it is often used as a secret weapon by those who know the technique. This book illuminates the subject with some of the most striking applications in analysis, topology, functional analysis, probability and stochastic analysis, as well as applications in economics and combinatorial number theory. The first chapter is designed to facilitate the beginner in learning this technique by starting with calculus and basic real analysis. The second chapter provides the reader with the most important tools of nonstandard analysis: the transfer principle, Keisler s internal definition principle, the spill-over principle, and saturation. The remaining chapters of the book study different fields for applications; each begins with a gentle introduction before then exploring solutions to open problems. All chapters within this second edition have been reworked and updated, with several completely new chapters on compactifications and number theory. Nonstandard Analysis for the Working Mathematician will be accessible to both experts and non-experts, and will ultimately provide many new and helpful insights into the enterprise of mathematics

The effective application of probabilistic reasoning in the study of problems in diverse areas is one of the most exciting recent developments in Mathematics. Probabilistic methods turned out to be very powerful in Discrete Mathematics, Analysis, Number Theory and Theoretical Computer Science. The meeting was dedicated to recent developments in these areas, focusing on the investigation of combinatorial problems for random sets and probabilistic methods, on the study of of questions in percolation, on the design and analysis of randomized algorithms and derandomization techniques, and on applications of probabilistic ideas in the study of questions in Combinatorial Number Theory and in Combinatorial Geometry.

We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number [Formula: see text] is the sum of at most three natural numbers whose base-[Formula: see text] representation has an equal number of [Formula: see text]’s and [Formula: see text]’s.

We consider a finite set of lattice points and their convex hull. The author previously gave a geometric proof that the sumsets of these lattice points take over the central regions of dilated convex hulls, thus revealing an interesting connection between additive number theory and geometry. In this paper, we will see an algebraic proof of this fact when the convex hull of points is a simplex, exploring the connection between additive number theory and geometry further.

Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plunnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.