DOCSLIB.ORG

Effective Schnirelmann's Method for O-Regular

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Effective Schnirelmann's Method for O-Regular EFFECTIVE SCHNIRELMANN'S METHOD FOR O-REGULAR SEQUENCES CHRISTIAN TAFULA´ Abstract. In this paper we introduce the notion of pre-basis, which is a sequence such that some of its h-fold sumsets has positive lower asymptotic density, the least such h being its pre-order. Schnirelmann's classical theory of sumsets is reinterpreted as characterizing additive bases in terms of pre-bases, and estimates for the order in terms of the pre-order are derived from the deep theorems of Mann and Kneser. Under certain regularity assumptions on the representation functions, we then derive estimates to the pre-order. This is achieved by studying sequences A = fa0 < a1 < a2 < : : :g ⊆ N for which A(2x) = O(A(x)) and a2n = O(an), and ends up providing a small shortcut to the proofs of the Schnirelmann-Goldbach theorem and Linnik's elementary solution of Waring's problem. 1. Introduction Given A = fa0 < a1 < a2 < : : :g ⊆ N an infinite sequence and h ≥ 2 an Ph integer, the h-fold sumset hA is the set f i=1 ki : k1; : : : ; kh 2 A g. We say that A is an additive basis (resp. additive asymptotic basis) when there is h ≥ 2 such that N n hA is empty (resp. finite), and the least such h is called the order (resp. asymptotic order) of A , denoted by O(A ) (resp. O(A )). The representation functions rA ;h(n) and sA ;h(x) count the number of solutions of k1 + k2 + ::: + kh = n and k1 + k2 + ::: + kh ≤ x resp., where h ≥ 1 is a fixed integer and ki 2 A , considering permutations in the sense of the formal series: h ! P ah X X z X (1.1) za = r (n)zn; a2A = s (n)zn: A ;h 1 − z A ;h a2A n≥0 n≥0 Certain regularity properties on representation functions are essential to the elementary arguments showing that primes (Schnirelmann-Goldbach theorem, cf. Chapter 7 of Nathanson [4]) and sequences generated by poly- nomials (Generalized Waring's problem, cf. Chapters 11 & 12 of Nathanson [6]) are additive bases. In this paper we investigate some of these regularity properties in a general setting. Denoting by A(x) the quantity jA \ [0; x]j, we study three types of sequences: the O-regular sequences (OR), which satisfy A(2x) = O(A(x)); the positively increasing sequences (PI), which satisfy a2n = O(an); and the 2010 Mathematics Subject Classification. Primary 11B13, 11B34. Key words and phrases. additive bases, representation functions, O-regular variation, Schnirelmann's theorem, Schnirelmann's method. 1 2 CHRISTIAN TAFULA´ O-regular plus sequences (OR+), which are sequences being both OR and PI. The first part of this work is dedicated to motivate the study of these sequences by the asymptotic behavior of their representation functions. In the second part we introduce the notion of additive pre-bases. Given a sequence A ⊆ N, consider the following densities: Schnirelmann density: σ(A ) := inf (A(n) − A(0))=n; n≥1 Asymptotic density: d(A ) := lim A(x)=x; x!+1 Power density: w(A ) := lim log(A(x))= log(x); x!+1 the last two having lower (lim inf) and upper (lim sup) forms, signalized by a lower or upper bar, resp. (e.g. w, d). We define the Schnirelmann pre-order and pre-order of A as, respectively, Oe(A ) := minfh ≥ 1 : σ(hA ) > 0g; Oe(A ) := minfh ≥ 1 : d(hA ) > 0g: Thus pre-bases are sequences A with finite pre-order. The motivation for this concept comes from Schnirelmann's characterization of additive bases (see Theorem 3.1), which basically states: • B ⊆ N is a basis () Oe(B) < +1 and f0; 1g ⊆ B; • A ⊆ N is an asymptotic basis () Oe(A ) < +1 and A is not contained in an arithmetic progression with common difference r ≥ 2. Our main result is a combination of effective bounds for the order of a sequence in terms of its pre-order and a regularity condition on representation functions that ensures a sequence is a pre-basis. This last ingredient relates to what is loosely referred to in the literature as Schnirelmann's method. In order to state it, say a pre-basis A is stable when every A 0 ⊆ A with A0(x) = Θ(A(x)) is also a pre-basis. Moreover, say it is uniformly stable when there is h ≥ 1 such that Oe(A 0) ≤ h for every such A 0 ⊆ A . Main Theorem (Effective Schnirelmann's method). Let A ⊆ N be an OR+ sequence. If for some h ≥ 1 we have A(n)h X r (n) ξ(n); where ξ(n)2 = O(x); A ;h n n≤x then A is a uniformly stable pre-basis, with Oe(A 0) ≤ h for every subsequence A 0 ⊆ A with A0(x) = Θ(A(x)). Furthermore: • If f0; 1g ⊆ A , then the order of A is at most dσ(hA )−1eh; • If A is not contained in an arithmetic progression, then the asymp- totic order of is at most dd(h )−1 + 1 ( 1−d(hA ) )eh + bd(h )ch. A A h d(hA ) A This statement is the combination of Corollary 3.4 with Theorem 3.1, and it constitutes a small shortcut to the proof of the Schnirelmann-Goldbach theorem and the elementary solution of the generalized Waring's problem, covering their relatively easy parts. EFFECTIVE SCHNIRELMANN'S METHOD FOR O-REGULAR SEQUENCES 3 Notation. A real function f is asymptotically defined when f is well-defined in [α; +1) for some α ≥ 0. Whenever we write an \α" not specified at the context, this is the meaning that will be implicitly implied. Our use of asymptotic notation (Θ; ; O; ; o; ∼) is standard, with the addition of \≺", which has the same meaning as small-o (i.e. f ≺ g () f = o(g))1. 2. O-regularity in sequences We start with some generalities. Given a sequence A ⊆ N, let n 2 N and h ≥ 2 an integer. Then, from the formal series (1.1) one can immediately deduce the following recursive formulas: X rA ;h(n) = rA ;h−`(k)rA ;`(n − k) k≤n (2.1) X X sA ;h(n) = rA ;h−`(k)sA ;`(n − k) = sA ;h−`(k)rA ;`(n − k) k≤n k≤n For purposes of induction the case ` = 1 is usually enough, and we denote 1 rA ;1(n) simply by A (n). The following proposition motivates our study of O-regularity in sequences. Proposition 2.1. For every sequence A ⊆ N, sA ;h+1(x) sA ;h(x); 8h ≥ 1: Proof. Choose some 0 < " < 1 and define: 1−" L1(x) := A(x) ; 1−" Lh+1(x) := A(Lh(x)) ; 8h ≥ 1: We will use induction to show that, for all h ≥ 1, sA ;h+1(x) (2.2) sA ;h(x) : A(Lh(x)) As Lh+1(x) ≺ Lh(x) and Lh(x) 1 for all h, it will imply the statement of our proposition. Let us start with h = 1. Given that X 1 sA ;1(x) − sA ;1(x − L1(x)) = A (n) ≺ sA ;1(x); x−L1(x)<n≤x it follows that sA ;1(x) ∼ sA ;1(x − L1(x)). Hence, by the recursive formulas in (2.1), X 1 sA ;2(n) = sA ;1(n − k) A (k) k≤n X 1 sA ;1(n − k) A (k) k≤L1(n) sA ;1(n − L1(n)) · A(L1(n)) sA ;1(n) · A(L1(n)); thus (2.2) holds for h = 1. 1This is one of G. H. Hardy's asymptotic symbols. Despite being a little old-fashioned, we think it provides a nice counterpart for small-o, in the same way \" does for Big-O. 4 CHRISTIAN TAFULA´ For the induction step, note that for every h ≥ 1 we have X 1 (2.3) rA ;h+1(n) = rA ;h(k) A (n − k) ≤ sA ;h(n) k≤n Taking h > 1 and assuming (2.2) valid for h − 1, we have, by (2.3), X sA ;h(x) − sA ;h(x − Lh(x)) = rA ;h(n) x−Lh(x)<n≤x X ≤ sA ;h−1(n) x−Lh(x)<n≤x Lh(x)sA ;h−1(x) sA ;h(x) Lh(x) ≺ sA ;h(x); A(Lh−1(x)) then it follows that sA ;h(x) ∼ sA ;h(x − Lh(x)). Hence: X 1 sA ;h+1(n) = sA ;h(n − k) A (k) k≤n X 1 sA ;h(n − k) A (k) k≤Lh(n) sA ;h(n − Lh(n)) · A(Lh(n)) sA ;h(n) · A(Lh(n)); thus (2.2) holds for all h. This is a quite general estimate, and with the right restraints on the growth of A(x) we can actually say substantially more. 2.1. OR sequences. Let us introduce some bits of regular variation theory. An extensive treatment on this topic can be found in Bingham, Goldie & Teugels [1]. We will only need the theory from Chapters 2. Take f an asymptotically defined positive real function. We say f is • Slowly varying if f(λx) ∼ f(x) for all λ > 0; • Regularly varying if f(λx) ∼ λρf(x), for all λ > 0 and some ρ 2 R; • O-regularly varying if f(λx) f(x) for all λ > 0. The generality of these definitions lies on Karamata's characterization theorem2, which states that if f(λx) ∼ g(λ)f(x) with g(λ) 2 (0; +1) for all λ > 0 and f is measurable, then f is regularly varying, i.e. there is ρ 2 R such that g(λ) ≡ λρ.
Load more

Recommended publications
  • 13 Jan 2018 Waring's Theorem for Binary Powers
    Waring’s Theorem for Binary Powers Daniel M. Kane∗ Mathematics University of California, San Diego 9500 Gilman Drive #0404 La Jolla, CA 92093-0404 USA [email protected] Carlo Sanna Dipartimento di Matematica “Giuseppe Peano” Universit`adegli Studi di Torino Via Carlo Alberto 10 10123 Torino Italy [email protected] Jeffrey Shallit† School of Computer Science University of Waterloo Waterloo, ON N2L 3G1 Canada [email protected] arXiv:1801.04483v1 [math.NT] 13 Jan 2018 January 16, 2018 Abstract A natural number is a binary k’th power if its binary representation consists of k consecutive identical blocks. We prove an analogue of Waring’s theorem for sums of binary k’th powers. More precisely, we show that for each integer k ≥ 2, there exists a ∗Supported by NSF Award CCF-1553288 (CAREER) and a Sloan Research Fellowship. †Supported by NSERC Discovery Grant #105829/2013. 1 k positive integer W (k) such that every sufficiently large multiple of Ek := gcd(2 − 1, k) is the sum of at most W (k) binary k’th powers. (The hypothesis of being a multiple of Ek cannot be omitted, since we show that the gcd of the binary k’th powers is Ek.) Also, we explain how our results can be extended to arbitrary integer bases b> 2. 1 Introduction Let N = {0, 1, 2,...} be the natural numbers and let S ⊆ N. The principal problem of additive number theory is to determine whether every integer N (resp., every sufficiently large integer N) can be represented as the sum of some constant number of elements of S, not necessarily distinct, where the constant does not depend on N.
    [Show full text]
  • Melvyn B. Nathanson 1 X2a X   the Shnirel’Man Density of a Is Paul Erd˝Osand Additive Number Theory A(N) Additive Bases (A) Inf
    [Sh] Pablo Shmerkin, On the exceptional set for abso- new density for a set of integers that is exactly the lute continuity of Bernoulli convolutions, Geom. right density for the investigation of additive bases. Funct. Anal. 24 (2014), 946–958. (For a survey of the classical bases in additive [So] Boris Solomyak, On the random series λn (an ± number theory, see Nathanson [28].) Erd˝osproblem), Ann. of Math. (2), 142 P(3) (1955), 611–625. [W] Hermann Weyl, Über die Gleichverteilung von Shnirel’man Density and Essential Components Zahlen mod. Eins, Math. Annalen 77 (1916), 313– The counting function A(x) of a set A of nonnegative 352, Satz 21. integers counts the number of positive integers in [Z] Antoni Zygmund, Trigonometric Series. I, II, A that do not exceed x, that is, Cambridge Univ. Press, 1959. A(x) 1. = a A Melvyn B. Nathanson 1 X2a x The Shnirel’man density of A is Paul Erd˝osand Additive Number Theory A(n) Additive Bases σ(A) inf . = n 1,2,... n Paul Erd˝os,while he was still in his twenties, = The sum of the sets A and B is the set A B wrote a series of extraordinarily beautiful papers + = a b : a A and b B . Shnirel’man proved the in additive and combinatorial number theory. The { + 2 2 } fundamental sumset inequality: key concept is additive basis. Let be a set of nonnegative integers, let be σ(A B) σ(A) σ(B) σ(A)σ(B). A h + ≥ + − a positive integer, and let hA denote the set of This implies that if σ(A) > 0, then A is a basis of integers that can be represented as the sum of order h for some h.
    [Show full text]
  • Waring's Problem
    MATHEMATICS MASTER’STHESIS WARING’SPROBLEM JANNESUOMALAINEN HELSINKI 2016 UNIVERSITYOFHELSINKI HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI Tiedekunta/Osasto — Fakultet/Sektion — Faculty Laitos — Institution — Department Faculty of Science Department of Mathematics and Statistics Tekijä — Författare — Author Janne Suomalainen Työn nimi — Arbetets titel — Title Waring’s Problem Oppiaine — Läroämne — Subject Mathematics Työn laji — Arbetets art — Level Aika — Datum — Month and year Sivumäärä — Sidoantal — Number of pages Master’s Thesis 9/2016 36 p. Tiivistelmä — Referat — Abstract Waring’s problem is one of the two classical problems in additive number theory, the other being Goldbach’s conjecture. The aims of this thesis are to provide an elementary, purely arithmetic solution of the Waring problem, to survey its vast history and to outline a few variations to it. Additive number theory studies the patterns and properties, which arise when integers or sets of integers are added. The theory saw a new surge after 1770, just before Lagrange’s celebrated proof of the four-square theorem, when a British mathematician, Lucasian professor Edward Waring made the profound statement nowadays dubbed as Waring’s problem: for all integers n greater than one, there exists a finite integer s such that every positive integer is the sum of s nth powers of non- negative integers. Ever since, the problem has been taken up by many mathematicians and state of the art techniques have been developed — to the point that Waring’s problem, in a general sense, can be considered almost completely solved. The first section of the thesis works as an introduction to the problem. We give a profile of Edward Waring, state the problem both in its original form and using present-day language, and take a broad look over the history of the problem.
    [Show full text]
  • COMBINATORIAL NUMBER THEORY 1St Lecture Perhaps, You Have
    COMBINATORIAL NUMBER THEORY KATALIN GYARMATI 1st Lecture Perhaps, you have heard about Hilbert's problems. Hilbert proposed 23 problems, and ten of them were presented at the Second Interna- tional Congress in Paris on August 8th, 1900. Hilbert's problems lead to a significant advance in mathematics. The 10th problem of his was the following: Does there exist a universal algorithm (finite) for solving Diophantine equations? An equation is called Diophantine if we are looking for its integer solutions. Today we still study Diophantine equations. Could you tell me some examples of Diophantine equations? 5x + 12y = 1 This is a linear Diophantine equation. xn + yn = zn Fermat's \last theorem" For n = 2: x2 + y2 = z2 Pythagorean triple x4 + y4 + z4 = w4 Euler conjectured this equation has only trivial solutions. Elkies proved this is not true in 1988. Typical questions about Diophantine equations: 1. Are there any solutions? 2. Are there any further solutions beyond than the ones found easily? 3. Are there finitely or infinitely many solutions? 4. Can one describe all the solutions? 5. Can one in practice determine a full list of solutions? About the first question: sometimes (if we are lucky) there is an easy way to show that there is no solution to the Diophantine equation at all. Examples: 1. x2 + y2 = 3z2 has no solution in + N : There exist infinitely many m 2 Z such that 1 2 KATALIN GYARMATI 2. x3 + y3 + z3 = m has no solution in Z. (Waring problem) Returning to Hilbert's 10th Problem: Does there exist a universal algorithm for solving all Diophantine equations? This was unsolved for a long period.
    [Show full text]
  • An Extension of the Erd\H {O} S-Tetali Theorem
    AN EXTENSION OF THE ERDOS–TETALI˝ THEOREM CHRISTIAN TAFULA´ Abstract. Given a sequence A = {a0 < a1 < a2 ...} ⊆ N, let rA ,h(n) denote the number of ways n can be written as the sum of h elements of A . Fixing h ≥ 2, we show that if f is a suitable real function (namely: locally integrable, O-regularly varying and of positive increase) satisfying x1/(h−1) x1/h log(x)1/h ≪ f(x) ≪ for some ε> 0, log(x)ε then there must exist A ⊆ N with |A ∩ [0, x]| = Θ(f(x)) for which rA ,h+ℓ(n)= Θ(f(n)h+ℓ/n) for all ℓ ≥ 0. Furthermore, for h = 2 this condition can be weakened to x1/2 log(x)1/2 ≪ f(x) ≪ x. The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann’s method. 1. Introduction Denote by N the set of natural numbers with 0. Given a sequence A = {a0 < A h a1 < a2 ...} ⊆ N and an integer h ≥ 2, the h-fold sumset h is the set { i=1 ki : k1,...,kh ∈ A }. We say that A is an additive basis when there is h ≥ 2 such that N \ hA is empty, and the least such h is the order of A . An h-basis is anP additive basis of order h. The representation functions rA ,h(n) and sA ,h(x) count the number of solutions of k1 + k2 + ..
    [Show full text]
  • Adding Prime Numbers
    Adding Prime Numbers University of Cambridge Faculty of Mathematics Department of Pure Mathematics and Mathematical Statistics Author Markus Schepke BSc College Wolfson College Home address Robert-Beltz-Str. 49, 19059 Schwerin, Germany eMail [email protected] Adviser Professor Ben J. Green I declare that this essay is work done as part of the Part III Examination. I have read and understood the Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics, and have abided by it. This essay is the result of my own work, and except where explicitly stated otherwise, only includes material undertaken since the publication of the list of essay titles, and includes nothing which was performed in collaboration. No part of this essay has been submitted, or is concurrently being submitted, for any degree, diploma or similar qualification at any university or similar institution. Cambridge, November 20, 2011 Adding Prime Numbers University of Cambridge Faculty of Mathematics Department of Pure Mathematics and Mathematical Statistics Contents 1 Introduction 1 1.1 Historical Remarks and Outline . .1 1.2 Notations and Definitions . .1 2 Schnirelmann’s Theorem 3 2.1 Schnirelmann Density . .3 2.2 Proof of Schnirelmann’s Theorem . 11 2.3 Generalisations . 13 3 Chebyshev’s Theorem 16 3.1 Preliminary Notes . 17 3.2 Proof of Chebyshev’s Theorem . 20 4 The Selberg Sieve 22 4.1 Preliminary Notes . 22 4.2 Deduction of Schnirelmann’s Theorem . 28 4.3 Proof of the Selberg Sieve . 32 4.4 Applications to Twin Primes . 36 5 Waring’s Problem 40 5.1 Preliminary Notes .
    [Show full text]
  • Additive Number Theory the Classical Bases
    Graduate Texts in Mathematics Melvyn B. Nathanson Additive Number Theory The Classical Bases Springer Graduate Texts in Mathematics164 Editorial Board S. AxlerF.W. GehringP.R. Halmos Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo Graduate Texts in Mathematics ITAKEUTI/ZARING. Introduction to Axiomatic Set 35 WFRmFR. Banach Algebras and Several Theory. 2nd ed. Complex Variables. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 36 KELLEY/NAMIOKA El AL. Linear Topological 3 SCHAEFER. Topological Vector Spaces. Spaces. 4 HILTONISTAMMBACH. A Course in Homological 37 MONK. Mathematical Logic. Algebra. 38 GRAUERT/FRrrZSCHE. Severa! Complex 5 MAC LANE. Categories for the Working Variables. Mathematician. 39 ARVtsoN. An Invitation to C'-Algcbras. 6 HUGHES/PIPER. Projective Planes. 40 KEMENY/SNELL/KJ APP. Denumerable Markov 7 SFRRE. A Course in Arithmetic. Chains. 2nd ed. 8 TAKEUn2ARuNG. Axiomatic Set Theory. 41 APOSTOL. Modular Functions and Dirichlet 9 HUMPHREYS. Introduction to Lie Algebras and Series in Number Theory. 2nd ed. Representation Theory. 42 SERRE. Linear Representations of Finite Groups. 10 COHEN. A Course in Simple Homotopy Theory. 43 GILLMAN/JERISON. Rings of Continuous I I CONWAY. Functions of One Complex Variable Functions. 1. 2nd ed. 44 KazmiG. Elementary Algebraic Geometry. 12 BEMs. Advanced Mathematical Analysis. 45 LOEvE. Probability Theory 1. 4th ed. 13 ANDERsoN/FuLLEtt. Rings and Categories of 46 LoEva. Probability Theory 11. 4th ed. Modules. 2nd ed. 47 Mots.. Geometric Topology in Dimensions 2 14 GOLUBrrSKY/GUILLEMIN. Stable Mappings and and 3. Their Singularities. 48 SACHS/WI. General Relativity for 15 BERBERIAN. Lectures in Functional Analysis Mathematicians.
    [Show full text]
  • On Additive Bases in Infinite Abelian Semigroups 3
    ON ADDITIVE BASES IN INFINITE ABELIAN SEMIGROUPS PIERRE-YVES BIENVENU, BENJAMIN GIRARD, AND THAI´ HOANG` LEˆ Abstract. Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups. We show that, for every such group T , the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality k contained in an additive basis of order at most h can be bounded in terms of h and k alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon N. Also, using invariant means, we address a classical problem, initiated by Erd˝os and Graham and then generalized by Nash and Nathanson both in the case of N, of estimating the maximal order XT (h, k) that a basis of cocardinality k contained in an additive basis of order at most 2k+1 h can have. Among other results, we prove that XT (h, k)= O(h ) for every integer k > 1. This result is new even in the case where k = 1. Besides the maximal order XT (h, k), the typical order ST (h, k) is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in N and in abelian groups. 1. Introduction Let (T, +) be an abelian semigroup. If A, B are two subsets of T whose symmetric difference is finite, we write A ∼ B.
    [Show full text]
  • Andrew Granville Introduction for A, B Subsets of an Additive Group Z, We
    ADDITIVE COMBINATORICS (WINTER 2005) Andrew Granville Introduction For A,B subsets of an additive group Z, we define A + B to be the sumset a + b : a A,b B , and kA to be the k-fold sum A + A + + A of A. We also{ let A B ∈= a ∈b : a } A,b B and b+A = b +A for a single··· element set b ,a translate of−A. Note{ − that A∈ A is∈ not} 0 unless A {=} 1. We let k A = ka : a { }A , a dilate of A. There are many− obvious properties| of| “+” that can be⋄ checked{ like commutativity,∈ } associativity and the distributive law A +(B C)=(A + B) (A + C). ∪ ∪ Prove that k A kA and classify when they are equal. Prove that b + A = A . Show that A ⋄ ⊆ | | | | | | ≤ A + B A B . Describe the situations when we get equality. Improve this last upper bound for A + A | |≤| || | | | and for A A . | − | We shall be most interested in understanding the size and structure of sumsets which are subsets of the integers Z, often working with Z≥m (where A≥m denotes the integers in A that are m), or of Z/NZ for some positive integer N. That every integer is the sum of four squares≥ of integers can be written down as 4Z2 = Z (where here Z2 denotes the squares of the integers); if h(A 0 ) Z≥m for some m then we say that A is a basis of order h for the integers, and thus∪{Z2}is⊇ a basis of order 4.
    [Show full text]
  • Reflections on Paul Erd˝Os on His Birth Centenary
    Reflections on Paul Erd˝os on His Birth Centenary Krishnaswami Alladi and Steven Krantz, Coordinating Editors This is Part I of a two-part feature on Paul Erd˝osfollowing his centennial. There are eleven articles by leading experts who have reflected on the remarkable life, contributions, and influence of this towering figure of twentieth century mathematics. Here in Part I we have contributions from Krishnaswami Alladi and Steven Krantz, László Lovász and Vera T. Sós, Ronald Graham and Joel Spencer, Jean-Pierre Kahane, and Mel Nathanson. Part II will contain articles by Noga Alon, Dan Goldston, András Sárközy, József Szabados, Gérald Tenenbaum, and Stephan Garcia and Amy Shoemaker. Krishnaswami Alladi and Steven Krantz One of the Most Influential Mathematicians of Our Time The 100th birth anniversary of the great Hun- garian mathematician Paul Erd˝oswas celebrated in Budapest in July 2013 with an international conference that attracted about 750 participants. Erd˝oswas one of the most influential mathemati- cians of the twentieth century for a variety of reasons. He made fundamental and pioneering contributions in several fields of mathematics, such as number theory, combinatorics, graph theory, analysis, geometry, and set theory. He was perhaps the most prolific mathematician in history after Euler, with more than 1,500 papers, but what was most interesting about this was that more than half of these were joint papers and many of his collaborators were very young. Thus through the fundamental ideas in his papers and through these Courtesy of Harold Diamond Krishnaswami Alladi is professor of mathematics at the Uni- Paul Erd˝osworking in his office at the Hungarian versity of Florida.
    [Show full text]
  • Sumsets and Structure Imre Z. Ruzsa
    Sumsets and structure Imre Z. Ruzsa Alfred¶ Renyi¶ Institute of Mathematics, Budapest, Pf. 127, H-1364 Hungary, E-mail address: [email protected] Contents Foreword 1 Overview 3 Notation 4 Exercises and problems 4 Chapter 1. Cardinality inequalities 5 1. Introduction 5 2. PlÄunnecke's method 6 3. Magni¯cation and disjoint paths 8 4. Layered product 10 5. The independent addition graph 11 6. Di®erent summands 13 7. PlÄunnecke's inequality with a large subset 14 8. Sums and di®erences 15 9. Double and triple sums 18 10. A + B and A + 2B 20 11. On the noncommutative case 22 Chapter 2. Structure of sets with few sums 27 1. Introduction 27 2. Torsion groups 30 3. Freiman isomorphism and small models 32 4. Elements of Fourier analysis on groups 34 5. Bohr sets in sumsets 38 6. Some facts from the geometry of numbers 40 7. A generalized arithmetical progression in a Bohr set 41 8. Freiman's theorem 43 9. Arithmetic progressions in sets with small sumset 44 Chapter 3. Location and sumsets 47 1. Introduction 47 2. The Cauchy-Davenport inequality 47 3. Kneser's theorem 48 4. Sumsets and diameter, part 1 51 5. The impact function 52 6. Estimates for the impact function in one dimension 54 7. Multidimensional sets 56 8. Results using cardinality and dimension 58 9. The impact function and the hull volume 60 10. The impact volume 62 iii iv CONTENTS 11. Hovanskii's theorem 66 Chapter 4. Density 69 1. Asymptotic and Schnirelmann density 69 2. Schirelmann's inequality 71 3.
    [Show full text]
  • Arxiv:1401.7598V1 [Math.NT]
    PAUL ERDOS˝ AND ADDITIVE BASES MELVYN B. NATHANSON Abstract. This is a survey of some of Erd˝os’s work on bases in additive number theory. 1. Additive bases Paul Erd˝os, while he was still in his 20s, wrote a series of extraordinarily beautiful papers in additive and combinatorial number theory. The key concept is additive basis. Let A be a set of nonnegative integers, let h be a positive integer, and let hA denote the set of integers that can be represented as the sum of exactly h elements of A, with repetitions allowed. A central problem in additive number theory is to describe the sumset hA. The set A is called an additive basis of order h if every nonnegative integer can be represented as the sum of exactly h elements of A. For example, the set of squares is a basis of order 4 (Lagrange’s theorem) and the set of nonnegative cubes is a basis of order 9 (Wieferich’s theorem). The set A of nonnegative integers is an asymptotic basis of order h if hA contains every sufficiently large integer. For example, the set of squares is an asymptotic basis of order 4 but not of order 3. The set of nonnegative cubes is an asymptotic basis of order at most 7 (Linnik’s theorem), and, by considering congruences modulo 9, an asymptotic basis of order at least 4. The Goldbach conjecture implies that the set of primes is an asymptotic basis of order 3. The modern theory of additive number theory begins with the work of Lev Genrikhovich Shnirel’man (1905-1938).
    [Show full text]