On a Two–dimensional Analog of Szemerédi’s Theorem in Abelian Groups
نویسنده
چکیده
Let G be a finite Abelian group and A ⊆ G×G be a set of cardinality at least |G|/(log log |G|)c, where c > 0 is an absolute constant. We prove that A contains a triple {(k, m), (k + d, m), (k, m+ d)}, where d 6= 0. This theorem is a two-dimensional generalization of Szemerédi’s theorem on arithmetic progressions.
منابع مشابه
On non-normal non-abelian subgroups of finite groups
In this paper we prove that a finite group $G$ having at most three conjugacy classes of non-normal non-abelian proper subgroups is always solvable except for $Gcong{rm{A_5}}$, which extends Theorem 3.3 in [Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable, Acta Math. Sinica (English Series) 27 (2011) 891--896.]. Moreover, we s...
متن کاملOn continuous cohomology of locally compact Abelian groups and bilinear maps
Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...
متن کاملAddendum to: "Infinite-dimensional versions of the primary, cyclic and Jordan decompositions", by M. Radjabalipour
In his paper mentioned in the title, which appears in the same issue of this journal, Mehdi Radjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminolo...
متن کاملTriple factorization of non-abelian groups by two maximal subgroups
The triple factorization of a group $G$ has been studied recently showing that $G=ABA$ for some proper subgroups $A$ and $B$ of $G$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups $D_{2n}$ and $PSL(2,2^{n})$...
متن کاملNew Bounds for Szemerédi’s Theorem, I: Progressions of Length 4 in Finite Field Geometries
Let k > 3 be an integer, and let G be a finite abelian group with |G| = N , where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality |A| of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that rk(Z/NZ) = ok(N). It is known, in fact, that the estimate rk(G) = ok(N) holds for all G. There have been ma...
متن کامل