The Circle Method
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The Circle Method Applied to Goldbach's Weak Conjecture
CARLETON UNIVERSITY SCHOOL OF MATHEMATICS AND STATISTICS HONOURS PROJECT TITLE: The Circle Method Applied to Goldbach’s Weak Conjecture AUTHOR: Michael Cayanan SUPERVISOR: Brandon Fodden DATE: August 27, 2019 Acknowledgements First and foremost, I would like to thank my supervisor Brandon Fodden not only for all of his advice and guidance throughout the writing of this paper, but also for being such a welcoming and understanding instructor during my first year of undergraduate study. I would also like to thank the second reader Kenneth Williams for graciously providing his time and personal insight into this paper. Of course, I am also very grateful for the countless professors and instructors who have offered me outlets to explore my curiosity and to cultivate my appreciation of mathematical abstraction through the courses that they have organized. I am eternally indebted to the love and continual support provided by my family and my dearest friends throughout my undergraduate journey. They were always able to provide assurance and uplift me during my worst and darkest moments, and words cannot describe how grateful I am to be surrounded with so much compassion and inspiration. This work is as much yours as it is mine, and I love you all 3000. THE CIRCLE METHOD APPLIED TO GOLDBACH'S WEAK CONJECTURE Michael Cayanan August 27, 2019 Abstract. Additive number theory is the branch of number theory that is devoted to the study of repre- sentations of integers subject to various arithmetic constraints. One of, if not, the most popular question that additive number theory asks arose in a letter written by Christian Goldbach to Leonhard Euler, fa- mously dubbed the Goldbach conjecture. -
Report on Diophantine Approximations Bulletin De La S
BULLETIN DE LA S. M. F. SERGE LANG Report on Diophantine approximations Bulletin de la S. M. F., tome 93 (1965), p. 177-192 <http://www.numdam.org/item?id=BSMF_1965__93__177_0> © Bulletin de la S. M. F., 1965, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. Soc. math. France^ 93, ig65, p. 177 a 192. REPORT ON DIOPHANTINE APPROXIMATIONS (*) ; SERGE LANG. The theory of transcendental numbers and diophantine approxi- mations has only few results, most of which now appear isolated. It is difficult, at the present stage of development, to extract from the lite- rature more than what seems a random collection of statements, and this causes a vicious circle : On the one hand, technical difficulties make it difficult to enter the subject, since some definite ultimate goal seems to be lacking. On the other hand, because there are few results, there is not too much evidence to make sweeping conjectures, which would enhance the attractiveness of the subject. With these limitations in mind, I have nevertheless attempted to break the vicious circle by imagining what would be an optimal situa- tion, and perhaps recklessly to give a coherent account of what the theory might turn out to be. -
Mathematical Tripos Part III Lecture Courses in 2020-2021
Mathematical Tripos Part III Lecture Courses in 2020-2021 Department of Pure Mathematics & Mathematical Statistics Department of Applied Mathematics & Theoretical Physics Notes and Disclaimers. • Students may attend any combination of lectures but should note that it is not possible to take two courses for examination that are lectured in the same timetable slot. There is no requirement that students study only courses offered by one Department. • The code in parentheses after each course name indicates the term of the course (M: Michaelmas; L: Lent; E: Easter), and the number of lectures in the course. Unless indicated otherwise, a 16 lecture course is equivalent to 2 credit units, while a 24 lecture course is equivalent to 3 credit units. Please note that certain courses are non-examinable, and are indicated as such after the title. Some of these courses may be the basis for Part III essays. • At the start of some sections there is a paragraph indicating the desirable previous knowledge for courses in that section. On the one hand, such paragraphs are not exhaustive, whilst on the other, not all courses require all the pre-requisite material indicated. However you are strongly recommended to read up on the material with which you are unfamiliar if you intend to take a significant number of courses from a particular section. • The courses described in this document apply only for the academic year 2020-21. Details for subsequent years are often broadly similar, but not necessarily identical. The courses evolve from year to year. • Please note that while an attempt has been made to ensure that the outlines in this booklet are an accurate indication of the content of courses, the outlines do not constitute definitive syllabuses. -
Additive Combinatorics with a View Towards Computer Science And
Additive Combinatorics with a view towards Computer Science and Cryptography An Exposition Khodakhast Bibak Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario, Canada N2L 3G1 [email protected] October 25, 2012 Abstract Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other arXiv:1108.3790v9 [math.CO] 25 Oct 2012 branches; all deeply cross-fertilize each other, holding great promise for all of them! In this exposition, we attempt to provide an overview of some breakthroughs in this field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography. 1 Introduction Additive combinatorics is a compelling and fast growing area of research in mathematical sciences, and the goal of this paper is to survey some of the recent developments and notable accomplishments of the field, focusing on both pure results and applications with a view towards computer science and cryptography. See [321] for a book on additive combinatorics, [237, 238] for two books on additive number theory, and [330, 339] for two surveys on additive 1 combinatorics. -
On Some Conjectures in Additive Number Theory Huixi Li Clemson University, [email protected]
Clemson University TigerPrints All Dissertations Dissertations 8-2018 On Some Conjectures in Additive Number Theory Huixi Li Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Recommended Citation Li, Huixi, "On Some Conjectures in Additive Number Theory" (2018). All Dissertations. 2209. https://tigerprints.clemson.edu/all_dissertations/2209 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. On Some Conjectures in Additive Number Theory A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Mathematics by Huixi Li August 2018 Accepted by: Dr. Jim Brown, Committee Chair Dr. Michael Burr Dr. Kevin James Dr. Hui Xue Abstract In the first part of the thesis we prove that every sufficiently large odd integer can be written as a sum of a prime and 2 times a product of at most two distinct odd primes. Together with Chen's theorem and Ross's observation, we know every sufficiently large integer can be written as a sum of a prime and a square-free number with at most three prime divisors, which improves a theorem by Estermann that every sufficiently large integer can be written as a sum of a prime and a square-free number. In the second part of the thesis we prove some results that specialize to confirm some conjec- tures of Sun, which are related to Fermat's theorem on sums of two squares and other representations of primes in arithmetic progressions that can be represented by quadratic forms. -
On the Inverse Erd˝Os-Heilbronn Problem For
ON THE INVERSE ERDOS-HEILBRONN} PROBLEM FOR RESTRICTED SET ADDITION IN FINITE GROUPS Suren M. Jayasuriyay Department of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA [email protected] Steven D. Reichy Department of Mathematics, The University of Pittsburgh, Pittsburgh, Pennsylvania, 15260, USA [email protected] Jeffrey P. Wheeler Department of Mathematics, The University of Pittsburgh, Pittsburgh, Pennsylvania, 15260, USA [email protected] Abstract We provide a survey of results concerning both the direct and inverse problems to the Cauchy-Davenport theorem and Erd}os-Heilbronnproblem in Additive Combina- torics. We formulate a conjecture concerning the inverse Erd}os-Heilbronn problem in nonabelian groups. We prove an inverse to the Dias da Silva-Hamidoune The- orem to Z=nZ where n is composite, and we generalize this result for nonabelian groups. 1. Introduction A basic object in additive combinatorics/additive number theory is the sumset of sets A and B: yThis work was started while S.M. Jayasuriya and S.D. Reich were undergraduates at the University of Pittsburgh in a directed study course supervised by Dr. Jeffrey P. Wheeler. 2010 Mathematics Subject Classification: Primary 11P99; Secondary 05E15, 20D60. Key words and phrases: Cauchy-Davenport Theorem, Erd}os-Heilbronn Problem, additive number theory, sumsets, restricted set addition, finite groups, inverse sumset results, critical pair. 1 Definition 1.1. [Sumset] A + B := fa + b j a 2 A; b 2 Bg: A simple example of a problem in Additive Number Theory is given two subsets A and B of a set of integers, what facts can we determine about sumset A + B? One such classic problem was a conjecture of Paul Erd}osand Hans Heilbronn [12], an open problem for over 30 years until proved in 1994. -
Diophantine-Geometry-Course.Pdf
DIOPHANTINE GEOMETRY { TAUGHT BY JOE SILVERMAN NOTES BY SHAMIL ASGARLI 1. Diophantine Equations Objective. Solve polynomial equations using integers (or rationals). Example. Linear equation: Solve ax + by = c where a; b; c 2 Z. It is a classical fact from elementary number theory that this equation has a solution (for x and y) if and only if gcd(a; b) j c. Example. Quadratic equation: x2 + y2 = z2. We are interested in non-zero solutions (x; y; z) 2 Z, i.e. (x; y; z) 6= (0; 0; 0). Since the equation is homogeneous, it is enough to understand the solutions of X2 + Y 2 = 1 where X; Y 2 Q (points on the unit circle). Anyways, the complete solution is known for this problem. WLOG gcd(x; y; z) = 1, x is odd and y is even. Then the solutions are given by x = s2 − t2, y = 2st, and z = s2 + t2. s2 − t2 Analogously, the solutions (X; Y ) 2 2 of X2 + Y 2 = 1 are parametrized by: X = Q s2 + t2 2st and Y = . s2 + t2 Example. Consider the equation y2 = x3 − 2. It turns out that (3; ±5) are the only solutions in Z2, while there are infinitely many solutions (x; y) 2 Q. Goal. Given f1; f2; :::; fk 2 Z[X1; :::; Xn]. For R = Z; Q, or any other ring. Let n V (R) = f(x1; x2; :::; xn) 2 R : fi(x1; :::; xn) = 0 for all ig Describe V (R). Two questions naturally arise. 1) Is V (R) = ;? This is undecidable for R = Z. 2) Is V (R) finite? 2 variables, 1 equation. -
Some Recent Progress on Diophantine Equations in Two-Variables
Some Recent Progress on Diophantine Equations In Two-Variables Minhyong Kim University of Warwick Rochester, May, 2021 I. Background: Arithmetic of Algebraic Curves Arithmetic of algebraic curves X : a smooth algebraic curve of genus g defined over Q. For example, given by a polynomial equation f (x; y) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)=2: Diophantine geometry studies the set X (Q) of rational solutions from a geometric point of view. Structure is quite different in the three cases: g = 0, spherical geometry (positive curvature); g = 1, flat geometry (zero curvature); g ≥ 2, hyperbolic geometry (negative curvature). Arithmetic of algebraic curves: g = 0; d = 2 Even now (after millennia of studying these problems), g = 0 is the only case that is completely understood. For g = 0, techniques reduce to class field theory and algebraic geometry: local-to-global methods, generation of solutions via sweeping lines, etc. Idea is to study Q-solutions by considering the geometry of solutions in various completions, the local fields R; Q2; Q3;:::; Q691;:::; Local-to-global methods Q2 RQ3 6 - ⊃ [ ⊂ ⊃ ⊂ - Q37 Q ⊂ ··· ⊃ \ - ? Q691 Q65537 Q282;589;933−1 Arithmetic of algebraic curves: g = 0 Local-to-global methods sometimes allow us to ‘globalise’. For example, 37x2 + 59y 2 − 67 = 0 has a Q-solution if and only if it has a solution in each of R; Q2; Q37; Q59; Q67; a criterion that can be effectively implemented. This is called the Hasse principle. If the existence of a solution is guaranteed, it can be found by an exhaustive search. -
491 Treatment Given Here for Product Measures in R2 Is Based on 2-Dimen
1964] BOOK REVIEWS 491 treatment given here for product measures in R2 is based on 2-dimen- sional interval functions, and seems to the reviewer to be distinctly more complicated than the usual treatment. XI. Derivatives and integrals. Vitali's covering theorem, differentiability of functions of bounded variation, etc., are treated. No one can quarrel with Professor Hildebrandt's intentions. Anal ysis should deal with concrete analytic entities, and one can easily lose sight of this in the morass of essential superior integrals, locally negligible sets, etc., of the most abstract treatments of integration theory. It is also true that many U.S. mathematicians [not merely graduate students] are unable to carry out manipulations that any competent Oxford undergraduate could do with ease. But the many avatars of Riemann-Stieltjes integration provide no remedy for this intellectual disease. The Daniell approach to integration, based upon Riemann-Stieltjes integrals for continuous functions, carries one quickly and easily to Lebesgue-Stieltjes integrals, and thence to all of the measure theory one can want, in any number of dimensions. Armed with this formidable weapon, one can then attack as many con crete problems as one wishes—everything from pointwise summabil- ity of Fourier series in many variables to partial differential equa tions. Professor Hildebrandt's book displays the author's mastery of the field, which he has known and contributed to nearly from its begin ning. In the reviewer's opinion, however, his book represents the history, and not the future, of integration theory. EDWIN HEWITT Diophantine geometry. By Serge Lang. Interscience Tracts in Pure and Applied Mathematics, No. -
Logic, Diophantine Geometry, and Transcendence Theory
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, January 2012, Pages 51–71 S 0273-0979(2011)01354-4 Article electronically published on October 24, 2011 COUNTING SPECIAL POINTS: LOGIC, DIOPHANTINE GEOMETRY, AND TRANSCENDENCE THEORY THOMAS SCANLON Abstract. We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier. 1. Introduction Over the past decade and a half, starting with Hrushovski’s proof of the func- tion field Mordell-Lang conjecture [12], some of the more refined theorems from model theory in the sense of mathematical logic have been applied to problems in diophantine geometry. In most of these cases, the technical results underlying the applications concern the model theory of fields considered with some additional distinguished structure, and the model theoretic ideas fuse algebraic model theory (the study of algebraic structures with a special emphasis on questions of defin- ability) and stability theory (the development of abstract notions of dimension, dependence, classification, etc.) for the purpose of analyzing the class of models of a theory. Over this period, there has been a parallel development of the model theory of theories more suited for real analysis carried out under the rubric of o-minimality, but this theory did not appear to have much to say about number theory. Some spectacular recent theorems demonstrate the error of this impression. In the paper [30], Pila presents an unconditional proof of a version of the so- called Andr´e-Oort conjecture about algebraic relations amongst the j-invariants of elliptic curves with complex multiplication using a novel technique that comes from model theory. -
Additive Combinatorics and Theoretical Computer Science∗
Additive Combinatorics and Theoretical Computer Science∗ Luca Trevisany May 18, 2009 Abstract Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications. In the past ten years, additive and arithmetic combinatorics have been extremely successful areas of mathematics, featuring a convergence of techniques from graph theory, analysis and ergodic theory. They have helped prove long-standing open questions in additive number theory, and they offer much promise of future progress. Techniques from additive and arithmetic combinatorics have found several applications in computer science too, to property testing, pseudorandomness, PCP constructions, lower bounds, and extractor constructions. Typically, whenever a technique from additive or arithmetic com- binatorics becomes understood by computer scientists, it finds some application. Considering that there is still a lot of additive and arithmetic combinatorics that computer scientists do not understand (and, the field being very active, even more will be developed in the near future), there seems to be much potential for future connections and applications. 1 Introduction Additive combinatorics differs from more classical work in extremal combinatorics for its study of subsets of the integers, or of more general abelian groups (rather than of graphs, hypergraphs, and set systems), and for asking questions that can be phrased in terms of linear equations (rather than being about cuts, intersections, subgraphs, and so on). -
Additive Number Theory Round
13th Annual Johns Hopkins Math Tournament Saturday, February 2, 2013 Explorations Unlimited Round − Additive Number Theory 1. Introduction Additive number theory has seen a spate of relatively recent results on very old, very hard, open problems; for example, the Green-Tao theorem settled a long-standing problem in 2004,1 and the seminal but older partial results by Chen on both the Goldbach Hypothesis and the Twin Primes problems remain current today.2;3 While these results are too complex to be summed up here, the older proofs presented here are worth revisiting to understand where the recent developments are coming from. We will work our way up to Dyson's Transform; this will give you all of the tools you will need to understand the proof of Mann's Theorem on your own. 2. The Basics: Core Definitions and Notation The set of natural numbers, N, is defined here as N = f0; 1; 2; 3; :::g. The set of real numbers is written as R. We use the notation a 2 A to denote that a is an element of a sequence called A. We use the notation jAj to denote the cardinality of a set A; that is, the number of elements in A. 3. Sequences Additive number theory, at its core, is about working with sequences. Here, we introduce addition of sequences, and later on, we'll work with the Dyson transform. At the heart of it all is the realization that having the mathematical tools to say things about several sequences at once is very useful.