It is independent of the related volume Additive Number Theory: The Classical Bases [961, which is a survey of the direct problems historically at the center of this subject. I have taught additive number theory at Southern Illinois University at Carbondale, Rutgers University-New Brunswick, and the Graduate Center of the City University of New York.
Direct and inverse problems
B={a-b:aEAandbEB}
Finite arithmetic progressions
Suppose h > 3, and suppose that the lower bound holds for the sum of all finite sets of integers of h - I. This follows from Theorem 1.6 and the fact that the set A is an arithmetic progression if and only if the normal form A(e') is an interval of integers.
An inverse problem for distinct summands
Since every interval A of length k satisfies condition (1.19), it follows from (1.20) that every k-term arithmetic progression also satisfies (1.19). The symmetry (1.15) implies that if A is an arithmetic progression when Ih ^ A I satisfies (1.19), then A is also an arithmetic progression when I(k - h)^AI satisfies (1.19).
A special case
Note that the It intervals are "right-shifting" in the sense that as a increases from 0 to h, the sequence of right endpoints of It intervals is strictly increasing and the sequence of left endpoints of Strictly increasing intervals.
UIt-[0,h(k-2)+t(b+1)]
This is impossible since a2 > 4. Let k > 4 and assume that the theorem holds for sets of cardinality k - 1. Simple inverse theorems follow. Therefore, we can assume that the numbers ak_3, ak_2 and ak_1 are in arithmetic progression.
Application: The number of sums and products
Application: Sumsets and powers of 2
Wirsing [128] proved that the Lebesgue measure of the set of real numbers corresponding to sets A such that A B + C for some B and C is zero. An important decomposition problem is the following: Are there infinite sets A and B of non-negative integers such that the sum set A + B and the set P of odd primes eventually coincide, i.e.
Exercises
A subsequent section (Nathanson [90]) will investigate the sums of infinite sets of integers in additive number theory. For example, it contains a profound and beautiful inverse theorem by Kneser [76] on the asymptotic density of sum sets, and an important recent improvement on this thanks to Bilu [10). For a comprehensive treatment of many of the key results of Waring's problem and Goldbach's conjecture, see Nathanson, Additive Number Theory: The Classical Bases [96].
Letk>3and
Let A be a finite subgroup of the abelian group G and let B be a finite subgroup of the abelian group H .
Fix r > 5, and let
- Addition in groups
- The e-transform
- The Cauchy-Davenport theorem
A basic tool to prove many results in additive number theory is the electronic transformation of an ordered pair (A, B) of nonempty subsets of an abelian group G. A basic result is the Cauchy-Davenport theorem, which gives a limit lower for the cardinality of the sum of two sets of congruence classes modulo a principal p.
- The Erd6s-Ginzburg-Ziv theorem
- Vosper's theorem
The multiplicative group of the nonzero elements of a finite field is cyclic, and thus, for any x E Fq,. In the case where n is a prime number p, the Erdos-Ginzburg-Ziv theorem (Theorem 2.5) is a corollary of the Chevalley warning theorem.
IA+BI=IAI+IBI - I
The inverse problem of addition in groups is to describe the structure of pairs of subsets (A, B) for which the cardinality of the sum A + B is small. For most pairs (A, B) in the abelian group G, the sum A + B will contain at least JAI + I BI elements. The group element d is called the total difference of the progression, and k is the length of the progression.
Using exponential sums and analytical methods, Freiman generalized Vosper's theorem for sum quantities of the form 2A = A + A in Z/pZ. If A and B are arithmetic progressions in G with the same common difference d, then there exist group elements a, b E G and positive integers k, I with k +1 < p such that.
IA+BI=IAI+IBI- 1
If A + B is an arithmetic progression, then A and B are arithmetic progressions with the same common difference. It follows that A is an arithmetic progression, and since the pair (A, B) is critical, the sets A and B are arithmetic progressions with the same common difference. If (A(e), B(e)) is any e-transformation of the critical pair (A, B), then Lemma 2.3 and the Cauchy-Davenport theorem imply that.
IAI+IBI - 1
A\X-leEA:B(e)-B)
- Application: The range of a diagonal form
- Exponential sums
- The Freiman-Vosper theorem
- Notes
- Exercises
- The Erdo"s-Heilbronn conjecture
- Vandermonde determinants
- Multidimensional ballot numbers
- A review of linear algebra
- Alternating products
- Erd6s-Heilbronn, concluded
Let m and x be integers such that m > 2, and let a - r + mZ be an element of the group Z/mZ of congruence classes modulo m. The second follows from the observation that the number of solutions of the equation is already +a2 - b in all, a2 EA and b E 2A IAI2 - V is. The next statement, which reduces the sums of congruence classes to sums of integers, is the important step in the proof of the theorem.
Let B(bo, b, , .., bh_ I) denote the number of ways m votes can be cast such that all kth ballot vectors are nonnegative and increasing. Let B(bo, b, , .., bh_,) denote the number of ways m votes can be cast such that all ballot vectors vA are nonnegative and strictly increasing. Let J(a, b) denote the number of paths from a to b that do not intersect any of the hyperplanes Hi.j.
A path from a to b is strictly increasing if and only if it intersects none of the hyperplanes H1 , and. If a is strictly increasing and a E Sh, a 7( id, then as is not strictly increasing, and therefore every path from as to b must intersect at least one of the hyperplanes Hi .j. Let £2 be the set of all intersecting paths starting at any of h.
The following result will be used later in the proof of the Erdo-Heilbronn conjecture.
VOAV1A...AVk_I EAV
- The polynomial method
- Erdo"s-Heilbronn via polynomials
- Notes
- Exercises
- Periodic subsets
- The addition theorem
The alternating product ^ h V is\ a vector space with a basis consisting of (h) wedge products of the form. In the following two sections we give a second proof of the Erd6s-Heilbronn conjecture that uses only elementary polynomial manipulations. From theorem 3.1, the coefficient of the monomial xo is the strict number of the ballot B(b0, bl..,bh- I).
It follows that the coefficient of the monomial x0A"-1 .xh" I -I in f' is exactly the same as the coefficient of this monomial in f, and this coefficient is. Nathanson [93] simplified the Dias da Silva-Hamidoune method by replacing the representation theory with simple properties of the ballot numbers. The polynomial proof of the Erd6s-Heilbronn conjecture is due to Alon, Nathanson and Ruzsa [2, 3].
We will use it to generalize the inverse theorem for sums of the form 2A (Theorem 1.16) to sums of the form A + B, where A and B are nonempty, finite sets of integers. Using properties (2.1) and (2.3) of the e-transformation, we conclude that there exists a proper subgroup H i G such that.
ICI-IC,I
CiUC2U...UCn,
C - C'UC,, Lemma 4.1 implies that either
For each bi E B we consider the collection of all pairs of finite subsets (Ai, Bi) of G such that.
IA+BI+IHI
Application: The sum of two sets of integers
Our goal in this section is to show that if 1 A+BI is "small", then A and B are "large" subsets of arithmetic progressions with the same common difference. This inverse theorem for the sum A + B is a generalization of theorem 1.16. bt_1 } be non-empty finite set of integers such that. We must show that there are at least e integers in A + B that lie in the same congruence classes modulo ak-1 as other integers in A + B.
Let denote the number of non-empty sets B, or equivalently the number of congruence classes modulo d containing at least one element of B. Let denote the number of congruence classes modulo d containing at least one element of B.
Application: Bases for finite and or -finite groups
2 JAI,
Notes
The proof of Kneser's theorem (76) for abelian groups in Section 4.2 follows Kemperman [74]. Mann [84] gives a condensed proof of this result. Vosper (Theorem 2.7) classified critical pairs into finite cyclic groups Z/pZ, where p is a prime number. Classification of critical pairs of subsets of any abelian group is an open problem.
Important partial results are due to Kemperman [74], who used Kneser's addition theorem for abelian groups to study this problem, and Hamidoune [65, 661, who used graph theory. There are a few results about critical pairs in nonabelian groups.. extended Kneser's theorem to certain special pairs of subgroups of nonabelian groups. Hamidoune [63) found a short proof of a theorem that includes the Brailovsky-Freiman result as a special case.
The results in Section 4.3 on the inverse theorem for sums of the form A + B were originally obtained by Freiman [52].
Exercises
It generalizes the result of Ji and Nathanson [73] for arbitrary a-finite abelian groups and of Cherly and Deshouillers [17] in the special case of the a-finite group Fq[x] of polynomials over a finite field. Let q be a power of a prime number p and let G - Fq[x] denote the ring of polynomials with coefficients in the finite field Fq. Let m > 3 and let G - [x] be an additive abelian group of polynomials with coefficients in the ring Z of integers modulo m.
Prove that there exists a basis A of order 2 for Z such that every integer has a unique representation as the sum of two elements of A.
Small sumsets and hyperplanes
There exist constants ko - ko(n, c) and so* - eo(n, c) > 0 such that, if A is a bounded subset of an n-dimensional Euclidean space V and if A satisfies. The following example, in the case h - 2, shows that the upper bound for c in Theorem 5.1 is the best possible.
Linearly independent hyperplanes
Hm are linearly independent hyperplanes. hm are independent and a dual set of vectors exists. Hm are linearly dependent hyperplanes. hm are dependent and there exist scalars a1, .. are not all zero such that.
Blocks
Sum of vectors in Euclidean space Let f and a be vectors in V. The reflection of a through f is a vector.
- Proof of the theorem
Block B(a') defines 2n face hyperplanes F1,. followed by this and so. from Lemma 5.11 it follows that.
IAnHI< eolAI 5 E 'IWI 5soIWI
Let n > 2, and let
- Notes
- Exercises
- Lattices and determinants
- xia1:0<x1
- Convex bodies and Minkowski's First Theorem
- Application: Sums of four squares
- Successive minima and Minkowski's second theoremsecond theorem
- Bases for sublattices
For example, in Section 6.3 we shall give a geometric proof of Lagrange's theorem that every non-negative integer is the sum of four squares. The abelian group generated by these n linearly independent vectors is the set of all sums of the form. The determinant of the lattice A, denoted it(A), plays a fundamental role in the geometry of numbers.
We will prove that det(A) is independent of the choice of basis for the lattice A. The basic parallelepiped of the lattice A with respect to the basis (a an) is a set. Fix a basis for lattice A and let F - F(A) be the basic parallelepiped of lattice A with respect to this basis.
We will use Minkowski's First Theorem to give a simple proof of Lagrange's famous theorem that every non-negative integer can be represented as the sum of four squares. The successive minima of the convex body K with respect to the lattice A are the real numbers X1, A2,.
