Structure of large incomplete sets in abelian groups

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 \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.

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.

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.

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|.

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.

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

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group $${\mathbb {Z}}_{p^e}$$ . They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with $$p \equiv 1 \pmod {4}$$ and e is odd with $$e>1$$ ; and all lengths that are $$0 \pmod {4}$$ when p is an odd prime with $$p \equiv 3 \pmod {4}$$ and e is odd with $$e>1.$$

Modular invariance property of association schemes is recalled in connection with our joint work with François Jaeger. Then we survey codes over F 2 discussing how codes, through their (various kinds of) weight enumerators, are related to (various kinds of) modular forms through polynomial invariants of certain finite group actions and theta series. Recently, not only codes over an arbitrary finite field but also codes over finite rings and finite abelian groups are considered and have been studied extensively. We show how the determination of the solutions of the modular invariance property of finite abelian groups (our joint work with Jaeger) is used to define the concept of Type II codes over arbitrary finite abelian groups. As an example of the usefulness of such Type II codes, we give an application to hermitian modular forms.

Progression-free sets in finite abelian groups

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

Recently, the concept of (normal) bent partitions, which are partitions of elementary abelian groups having similar properties to spreads, has been introduced by Anbar and Meidl. A large number of bent partitions, so-called generalized semifield spreads, can be obtained from semifields with certain properties. A strongly related concept, namely Latin square partial difference set packings (LP-packings) in finite abelian groups, has also been introduced recently by Jedwab and Li. The examples for LP-packings in an elementary abelian group are obtained from spreads. LP-packings yield bent partitions (not only for elementary abelian groups). In this paper, we first point out that conversely, generalized semifield spreads yield LP-packings. As a result, there is a huge amount of LP-packings in elementary abelian groups, other than spreads. With some examples from ternary bent functions, we then show that normal bent partitions and LP-packings are not the same concept. Finally, we extend the lifting procedure from spreads to LP-packings in nonelementary abelian groups to a lifting procedure from some generalized spreads to LP-packings in nonelementary abelian groups and in larger elementary abelian groups. This potentially yields bent partitions other than generalized semifield spreads.

The cross number of minimal zero-sum sequences in finite abelian groups