On the critical pair theory in Z/pZ (original) (raw)

On subsets of abelian groups with no 3-term arithmetic progression

Journal of Combinatorial Theory Series A

A short proof of the following result of Brown and Buhler is given: For any ϵ > 0 there exists n0 = n0(ϵ) such that if A is an abelian group of odd order |A| > n0 and B ⊆ A with |B| > ϵ|A|, then B must contain three distinct elements x, y, z satisfying x + y = 2z.

On Freiman's 3k-4 theorem

One of the many theorems Freiman proved, in the second half of the twentieth century, in the subject which later came to be known as "structure theory of set addition", was 'Freiman's 3k−43k-43k−4 theorem' for subsets of Z\ZZ. In this article we introduce concept of a new `structure' on finite subsets of integers. Sets with this structure are quite useful in additive number theory in some contexts. Also we give some criterion for subsets to posses this structure. Then this is used to establish an analog of Freiman's 3k−43k-43k−4 theorem for the groups ZtimesG,\Z \times G,ZtimesG, where GGG is any abelian group.

Large minimal sets which force long arithmetic progressions

Journal of Combinatorial Theory, Series A, 1986

A classic theorem of van der Waerden asserts that for any positive integer k, there is an integer W(k) with the property that if IV> W(k) and the set { 1,2,..., W} is partitioned into r classes C,, C?,..., C,, then some C, will always contain a k-term arithmetic progression. Let us abbreviate this assertion by saying that { 1, 2,..., W} arrows AP(k) (written { 1, 2,..., IV} + AP(k)). Further, we say that a set X crifically arrows AP(k) if:(i) X arrows AP(k); (ii) for any proper subset X' c X, x' does not arrow AP(k). The main result of this note shows that for any given k there exist arbitrarily large sets X which critically arrow AP(k).

On Sets with a Small Subset Sum

Combinatorics, Probability and Computing, 1999

Let A be a subset of an abelian group G. The subset sum of A is the set [sum ](A) = {[sum ]x∈T[mid ]T⊂A}. We prove the following result. Let S be a generating subset of an abelian group G such that 0∉S and 14[les ][mid ]S[mid ]. Then one of the following conditions holds.(i) [mid ][sum ](S)[mid ][ges ]min([mid ]G[mid ] −3, 3[mid ]S[mid ]−3).(ii) There is an x∈S such that S[setmn ]{x} generates a proper subgroup of order less than (3[mid ]S[mid ]−3)/2.As a consequence, we obtain the following open case of an old conjecture of Diderrich. Let q be a composite odd number and let G be an abelian group of order 3q. Let S be a subset of G with cardinality q+1. Then every element of G is the sum of some subset of S.

A structure theorem for small sumsets in nonabelian groups

European Journal of Combinatorics, 2013

Let G be an arbitrary finite group and let S and T be two subsets such that |S| ≥ 2, |T | ≥ 2, and |T S| ≤ |T | + |S| − 1 ≤ |G| − 2. We show that if |S| ≤ |G| − 4|G| 1/2 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS| ≤ |S| + |H| − 1 or |SH| ≤ |S| + |H| − 1. This extends to the nonabelian case classical reults for abelian groups. When we remove the hypothesis |S| ≤ |G| − 4|G| 1/2 we show the existence of counterexamples to the above characterization whose structure is described precisely.

A refinement of a conjecture of Gross, Kohnen, and Zagier

Contemporary mathematics, 2018

AMS: Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Arithmetic aspects of modular and Shimura varieties. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Rational points. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Applications to coding theory and cryptography. msc | Number theory-Discontinuous groups and automorphic forms-Automorphic forms on. msc | Number theory-Algebraic number theory: global fields-Class field theory. msc | Number theory-Zeta and L-functions: analytic theory-None of the above, but in this section. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Modular and Shimura varieties. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-L-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Curves of arbitrary genus or genus = 1 over global fields.

The Erdős–Szeméredi problem on sum set and product set

Annals of Mathematics, 2003

The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erdős-Szemerédi [E-S]. (see also [El], [T], and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A + A small and then deriving that the product set AA is large (using Freiman's structure theorem) (cf. [N-T], [Na3]). We follow the reverse route and prove that if |AA| < c|A|, then |A + A| > c |A| 2 (see Theorem 1). A quantitative version of this phenomenon combined with the Plünnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in [E-S] on the growth in k. If g(k) ≡ min{|A[1]| + |A{1}|} over all sets A ⊂ Z of cardinality |A| = k and where A[1] (respectively, A{1}) refers to the simple sum (resp., product) of elements of A. (See (0.6), (0.7).) It was conjectured in [E-S] that g(k) grows faster than any power of k for k → ∞. We will prove here that ln g(k) ∼ (ln k) 2 ln ln k (see Theorem 2) which is the main result of this paper.