Maximal sum-free sets in finite Abelian groups, III

Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x+ y = z, with x, y, z belonging to the set A. Let SF (G) the sel of all sum-free subets of G and σ(G) deonotes the number n(log 2 |SF (G)|). In this article we shall improve the error term in asymptotic formula of σ(G) obtained in [GR05]. The methods used are a slight refinement of methods of [GR05]. Let G be a finite abelian group of order n. A subset A of G is said to be sum-free if there is no solution of the equation x+ y = z, with x, y, z belonging to the set A. Let SF (G) denotes the set of all sum-free subsets of G. This article is motivated by the question of studying the cardinality of the set SF (G). Definition: (I) Let μ(G) denotes the density of a largest sum-free subset of G, so that any such subset has size μ(G)n. (II) Given a set B ⊂ G we say that (x, y, z) ∈ B3 is a Schur triple of the set B if x+ y = z. Observing that all subsets of a sum-free set are sum-free we have the obvious inequality |SF (G)| ≥ 2 (1) Let the symbol σ(G) denotes the number n(log2 |SF (G)|). Then from (1) it follows trivially that σ(G) ≥ μ(G). In this article we improve the results of Ben Green and Imre Ruzsa [GR05] and prove the following two results. The theorem 1 follows immediately from theorem 2 and a result from [GR05], namely theorem 6. The methods used to prove theorem 2 are a slight refinements of methods in [GR05].

Let G be a finite abelian group and A be a subset of G. We say that A is complete if every element of G can be represented as a sum of different elements of A. In this paper, we study the following question What is the structure of a large incomplete set? We show that such a set is essentially contained in a maximal subgroup. As a co-product, we obtain a new proof for several earlier results, including a new proof for Diderrich’s conjecture in large groups.

In this paper, we show that under some conditions the existence of a difference set in G implies the existence of another difference set with the same parameters in G′, where G and G′ are abelian groups of the same order. This explains why there are more difference sets in abelian groups of low exponent and high rank than in those of high exponent and low rank.

Minimal sumsets in finite solvable groups

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

Addition sets in a finite group

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

Let G≅C n 1 ⊕...⊕C n r be a finite and nontrivial abelian group with n 1 |n 2 |...|n r . A conjecture of Hamidoune says that if W=w 1 ·...·w n is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S|≥|W|+|G|-1≥|G|+1, the maximum multiplicity of S at most |W|, and σ(W)≡0mod|G|, then there exists a nontrivial subgroup H such that every element g∈H can be represented as a weighted subsequence sum of the form g=∑ n i=1w i s i , with s 1 ·...·s n a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W|≥1 2|G|.A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S|≥|G|+𝔻(G)-1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g+H such that all but at most |G/H|-2 terms of S are from g+H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S|≥|G|+𝔻(G)-1 can be relaxed to |S|≥|G|+d * (G), where d * (G)=∑ r i=1(n i -1). We also use this method to derive a variation on Hamidoune’s conjecture valid when at least d * (G) of the w i are relatively prime to |G|.

We show that, for a positive integer r, every minimal 1-saturating set in ${PG}(r-1,2)$ of size at least $\frac{11}{36}\,2^r+3$ either is a complete cap or can be obtained from a complete cap S by fixing some $s\in S$ and replacing every point $s'\in S\setminus\{s\}$ by the third point on the line through s and $s'$. Since, conversely, every set obtained in this way is a minimal 1-saturating set and the structure of large sum-free sets in an elementary abelian 2-group is known, this provides a complete description of large minimal 1-saturating sets. An algebraic restatement is as follows. Suppose that G is an elementary abelian 2-group and a subset $A\subseteq G\setminus\{0\}$ satisfies $A\cup2A=G$ and is minimal subject to this condition. If $|A|\ge\frac{11}{36}\,|G|+3$, then either A is a maximal sum-free set or there are a maximal sum-free set $S\subseteq G$ and an element $s\in S$ such that $A=\{s\}\cup\bigl(s+(S\setminus\{s\})\bigr)$. Our approach is based on characterizing those large sets A in elementary abelian 2-groups such that, for every proper subset B of A, the sumset $2B$ is a proper subset of $2A$.

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

We define the positive diameter of a finite group G with respect to a generating set A ⊆ G to be the smallest non-negative integer n such that every element of G can be written as a product of at most n elements of A. This invariant, which we denote by , can be interpreted as the diameter of the Cayley digraph induced by A on G.

Given an additive group G and nonempty subsets S, T of G, let S+T denote the set ﹛s + t | s ∊ S, t ∊ T﹜, S the complement of S in G and |S| the cardinality of S. We call S a sum-free set in G if (S+S) ⊆ S. If, in addition, |S| ≥ |T| for every sum-free set T in G, then we call S a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G.

Progression-free sets in finite abelian groups

Every finite abelian group is a subgroup of the additive group of a finite simple left brace