Freiman theorem in an arbitrary abelian group pdf

In a commutative ring the invertible elements, or units, form an abelian multiplicative group. We need more than this, because two different direct sums may be isomorphic. Znz, n prime, resemble sets of integers from the additive point of view up to freiman isomorphism. Ruzsa submitted on 10 may 2005 v1, last revised 7 feb 2006 this version, v2. The finite abelian group is just the torsion subgroup of g. A is contained in an interval of length at most ms. We prove that a kapproximate subgroup of an arbitrary torsionfree nilpotent group can be covered by a bounded number of cosets of a nilpotent subgroup of bounded rank, where the bounds are explicit and depend only on k.

Theorem 4 mann let s be a subset of an arbitrary abelian group g. The result can be seen as a nilpotent analogue to freiman s dimension lemma. Ka, then a is contained within a multidimensional arithmetic progression of dimension dk and size fka. Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. Chevalleys theorem yields a faithfully at algebraic group map g. Freimans theorem in an arbitrary abelian group journal. Its group of points can be proven to be commutative. Because we avoid appealing to freiman s structure theorem, we get a reasonable bound. Suppose that g is an arbitrary abelian group and a is any finite subset g.

Citeseerx freimans theorem in an arbitrary abelian group. A freimantype theorem for locally compact abelian groups. By the definitions, an abelian variety is a group variety. Then gis said to be a simple group if its only normal subgroups are 1and g. Small doubling in ordered nilpotent groups of class 2.

If the group ais abelian, then all subgroups are normal, and so. We begin by establishing a correspondence principle between approximate groups and locally compact local groups that allows us to recover many results recently established in a fundamental paper of hrushovski. Razborov september 16, 20 abstract if a is a nite subset of a free group with at least two noncommuting elements then ja a aj jaj 2 logjajo1. For c, and hence by the lefschetz principle for every algebraically closed field of characteristic zero, the torsion group of an abelian variety of dimension g is isomorphic to qz 2g. Freimans theorem in an arbitrary abelian group core. Freimans theorem in finite fields via extremal set theory. Freimans theorem in an arbitrary abelian group dialnet.

The result is then discussed in light of the falconer distance problem. A brief introduction to approximate groups the library at msri. Freimans theorem in an arbitrary abelian group green. If any abelian group g has order a multiple of p, then g must contain an element of order p. We use the freiman theorem in arithmetic combinatorics to show that if the fourier transform of certain measures satisfies sufficiently bad estimates, then the support of the measure possesses an additive structure. Ruzsa, title freiman s theorem in an arbitrary abelian group, year 2006. Find out more about the kindle personal document service. If a is a subset of some abelian group, then by an smodel for a we mean a pair a. Journal of the london mathematical society, issn 00246107, vol. Abstract a famous result of freiman describes the structure of finite sets a. Hot network questions have china and india more than doubled carbon emissions since 2000 while u. Freiman s theorem in an arbitrary abelian group authors. A famous result of freiman describes the structure of finite sets a.

This was generalised by green and ruzsa to arbitrary abelian groups, where the controlling object is now a coset progression. In another direction, the cauchydavenport theorem was generalized to arbitrary abelian groups by mann 2, p. Freimans theorem in an arbitrary abelian group request pdf. Let gbe a nite abelian group of order n, written additively. Hindmans theorem in abelian groups sung hyup lee advisor. Freiman s theorem, fourier transform and additive structure of measures. More generally, the same conclusion holds in an arbitrary virtually free group, unless a generates a.

The free abelian group on s can be explicitly identified as the free group fs modulo the subgroup generated by its commutators, fs, fs, i. The fundamental theorem of finite abelian groups states, in part. The m obius function is strongly orthogonal to nilsequences. The fourier transform and equations over finite abelian groups. A freiman type theorem for locally compact abelian groups. A famous result of freiman describes the structure of. Introduction it follows easily from the fundamental theorem of finitely generated abelian groups that every. Ka, then a is contained within a multidimensional arithmetic progressio. In other words, the free abelian group on s is the set of words that are distinguished only up to the order of letters. For example, the following theorem characterizes all groups with abelian sylow 2subgroup. In fact, the claim is true if k 1 because any group of prime order is a cyclic group, and in this case any nonidentity element will. Freiman 1 gave the following improvement of vospers theorem in the case when.

A result due to ben green and imre ruzsa generalized freimans theorem to arbitrary abelian groups. While studying subsets s of size k of ordered nilpotent groups of class 2 with the small doubling property, we shall often try to reduce the hypotheses to those of the following proposition. In this paper, we will prove malles conjecture for number. Every ring is an abelian group with respect to its addition operation. The fourier transform and equations over finite abelian groups an introduction to the method of trigonometric sums. Applying freiman s theorem in an arbitrary abelian group see 11 to the commensurate subset a of a of bounded doubling again transferring from standard analysis to the nonstandard analysis. Let g, group of class 2 and let s be a subset of g of size k. A product theorem in free groups university of chicago. Bibtex entry for this abstract preferred format for this. American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Here we prove an analogous statement valid for subsets of an arbitrary abelian group.