Freiman theorem in an arbitrary abelian group pdf

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. 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. In other words, the free abelian group on s is the set of words that are distinguished only up to the order of letters. 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. Freimans theorem in an arbitrary abelian group request pdf. 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. A famous result of freiman describes the structure of. Here we prove an analogous statement valid for subsets of an arbitrary abelian group. In a commutative ring the invertible elements, or units, form an abelian multiplicative group.

A is contained in an interval of length at most ms. Freiman s theorem asserts, roughly speaking, if that a finite set in a torsionfree abelian group has small doubling, then it can be efficiently contained in or controlled by a generalised arithmetic progression. Chevalleys theorem yields a faithfully at algebraic group map g. 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. The result, as well as its proof, build upon and improve the fourth authors previous work 22, who proved this result with a restriction on which primes could divide a.

By the definitions, an abelian variety is a group variety. Ka, then a is contained within a multidimensional arithmetic progression of dimens. The finite abelian group is just the torsion subgroup of g. A result due to ben green and imre ruzsa generalized freimans theorem to arbitrary abelian groups. In this paper, we will prove malles conjecture for number. The fundamental theorem of finite abelian groups states, in part. Abstract a famous result of freiman describes the structure of finite sets a. 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. Theorem 4 mann let s be a subset of an arbitrary abelian group g. 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. Ruzsa submitted on 10 may 2005 v1, last revised 7 feb 2006 this version, v2. This was generalised by green and ruzsa to arbitrary abelian groups, where the controlling object is now a coset progression. Freiman s theorem, fourier transform and additive structure of measures. 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.

Every ring is an abelian group with respect to its addition operation. Hindmans theorem in abelian groups sung hyup lee advisor. The number of homomorphic images of an abelian group. A freimantype theorem for locally compact abelian groups. The fourier transform and equations over finite abelian groups. Then there is a normal subgroup kand a normal subgroup hwith k6 h, such that khas odd order, hhas odd index, and hkis a direct product of an abelian 2 group and simple groups with.

Freimans theorem in an arbitrary abelian group journal. Freimans theorem in an arbitrary abelian group dialnet. Small doubling in ordered nilpotent groups of class 2. The result can be seen as a nilpotent analogue to freiman s dimension lemma. If any abelian group g has order a multiple of p, then g must contain an element of order p. In another direction, the cauchydavenport theorem was generalized to arbitrary abelian groups by mann 2, p. Freiman s theorem for arbitrary abelian groups theorem 1. The m obius function is strongly orthogonal to nilsequences. Citeseerx freimans theorem in an arbitrary abelian group.

Znz, n prime, resemble sets of integers from the additive point of view up to freiman isomorphism. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication. Freimans theorem in an arbitrary abelian group core. An attempted proof of cauchys theorem for abelian groups using composition series. If a is a subset of some abelian group, then by an smodel for a we mean a pair a. A brief introduction to approximate groups the library at msri. For example, the following theorem characterizes all groups with abelian sylow 2subgroup. Hot network questions have china and india more than doubled carbon emissions since 2000 while u. If the group ais abelian, then all subgroups are normal, and so. We need more than this, because two different direct sums may be isomorphic. More generally, the same conclusion holds in an arbitrary virtually free group, unless a generates a. A famous result of freiman describes the structure of finite sets a of integers with small doubling property. Ruzsa, title freiman s theorem in an arbitrary abelian group, year 2006. 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.

Ka, then a is contained within a multidimensional arithmetic progressio. 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. Bibtex entry for this abstract preferred format for this. Find out more about the kindle personal document service. Let gbe a nite abelian group of order n, written additively. They used an analogous notion to generalized arithmetic progressions, which. 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. Freimans theorem in an arbitrary abelian group green. The fourier transform and equations over finite abelian groups an introduction to the method of trigonometric sums. Freiman s theorem in an arbitrary abelian group authors. Freiman 1 gave the following improvement of vospers theorem in the case when. This may be viewed as a generalisation of the freiman ruzsa theorem on sets of small doubling in the integers to arbitrary groups. Ka, then a is contained within a multidimensional arithmetic progression of dimension dk and size fka.

Because we avoid appealing to freiman s structure theorem, we get a reasonable bound. A freiman type theorem for locally compact abelian groups. Journal of the london mathematical society, issn 00246107, vol. Then gis said to be a simple group if its only normal subgroups are 1and g. A product theorem in free groups university of chicago. Suppose that g is an arbitrary abelian group and a is any finite subset g. The result is then discussed in light of the falconer distance problem. Freimans theorem in finite fields via extremal set theory. A famous result of freiman describes the structure of finite sets a. Let g, group of class 2 and let s be a subset of g of size k. Its group of points can be proven to be commutative. Introduction it follows easily from the fundamental theorem of finitely generated abelian groups that every.