Measure theory and number theory?

by pivoxa15
Tags: measure, number, theory
 P: 2,267 How is measure theory associated with number theory, if at all. If they are connected, can anyone give a link?
 Sci Advisor P: 6,102 There is no connection except that both are branches of mathematics. Number theory is concerned with integers and their properties. Measure theory, in its most general form, is concerned with abstract sets and quantifiable properties.
 Sci Advisor P: 6,102 Post script: Number theory was studied by the ancient Greeks (Euclid et al). Measure theory development is around 100 years old.
 P: 2,267 Measure theory and number theory? There is analytical number theory. Analysis is related to measure theory and so is integration so there is an indirect link at least.
 P: 291 There is also algebraic number theory, but algebra and number theory are related as soon as you start learning about them. However, saying that that "...is concerned with abstract sets and quantifiable properties..." is not a good answer because there are certain topics in number theory that are related to numbers but they come from a totally different area, for example, modular forms from complex analysis being applied to number theory. I would agree with you that there is no connection but I do not know.
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 Quote by Kummer There is also algebraic number theory, but algebra and number theory are related as soon as you start learning about them.
I assume you are referring to ring theory in algebra.
 Sci Advisor HW Helper PF Gold P: 4,771 I was reading yesterday on wikipedia that the Haar measure had application in number theory. http://en.wikipedia.org/wiki/Haar_measure
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 Quote by pivoxa15 I assume you are referring to ring theory in algebra.
No. Field theory in particular over finite fields and rational numbers. And Galois theory are used a lot in number theory. That is probably the largest area of algebra that is used in number theory. And p-adic analysis.
P: 2,267
 Quote by Kummer No. Field theory in particular over finite fields and rational numbers. And Galois theory are used a lot in number theory. That is probably the largest area of algebra that is used in number theory. And p-adic analysis.
All fields are rings, but not conversely.

You said, '...algebra and number theory are related as soon as you start learning about them...'

I assume one does not learn Galois theory as the first exposure to algebra. One does rings and group theory before moving on to Galois theory which uses a combination of them.
Emeritus
PF Gold
P: 4,500
 Quote by pivoxa15 All fields are rings, but not conversely. You said, '...algebra and number theory are related as soon as you start learning about them...' I assume one does not learn Galois theory as the first exposure to algebra. One does rings and group theory before moving on to Galois theory which uses a combination of them.
You can do number theory using rings and groups. For example, the group of integers 1,...p-1 with multiplication mod p can be used to show for all m, m^p-1=1 mod p, a trivial result once you talk about the order of a group. I'm sure there are other things I can't remember off the top of my head
 Sci Advisor HW Helper P: 9,495 here is talk title from an international conference on probability nd number theory: "Asymptotic probability measures of Dedekind zeta-functions of non-Galois fields." see: http://www.math.tohoku.ac.jp/~hattori/pnt5.htm
P: 75
 Quote by pivoxa15 How is measure theory associated with number theory, if at all. If they are connected, can anyone give a link?
These two fields are actually quite intimately connected. Statements like those of mathman at the beginning of the thread are completely against the spirit of mathematics. I suggest that if you had this impression about the various fields of mathematics, you read Terence Tao's essay What is Good Mathematics?

Now, on to measure theory and number theory! So, the first thing to note is that people often associate measure theory with real analysis because the first thing anyone did with it was generalize the definition of integration, and most of the early and famous applications of measure theory were of purely analytic, or geometric-analytic, interest. But measure theory is simply concerned with measures, which are (intuitively speaking) a kind of description of size, and there's no reason that you wouldn't care about how big things are in number theory. In fact, number theory is the study of the structure of the integers, and a significant part of structure is how big things are.

So, let me briefly mention analytic number theory. Basically, measure theory is essential to the foundation of modern analysis. Have you heard of the Fourier transform? L^2 functions? These concepts are central to analytic number theory, and they are founded on measure-theoretic principles. But I don't think this really answers pivoxa's question. This isn't really an overlap between the fields. This is more like measure theory providing some rigor and clarity to things that were pretty much already there.

Therefore, let's consider a more interesting situation. Ergodic Theory is a sub-field of measure theory; it's essentially the study of measure-preserving transformations on probability spaces. This seems like analysis, probability theory, whatever. Right? So you might be a bit surprised to hear that some of the best known results about additive structure of the integers can so far be proven onlyl using ergodic-theoretic techniques.

In 1975 (?), Hillel Furstenberg opened the door to these developments with his ergodic-theoretic proof of Szemeredi's theorem, which states that any subset of the integers with positive density contains arithmetic progressions of arbitrary length. More recently (and more sensationally), Ben Green and Terence Tao combined the concepts from the four independent proofs of Szemeredi's theorem (Szemeredi's combinatorial proof, Furstenberg's ergodic proof, Gowers' Fourier-analytic proof, and Gowers' hypergraph proof) and were able to show that the primes contain arithmetic progressions of arbitrary length. This monumental achievement, which was a marvelous unification of four seemingly different fields of mathematics (graph theory, Fourier analysis, additive number theory, ergodic theory) was part of the reason for Terry Tao's Fields Medal.

Furstenberg's proof
A relevant post on Terry Tao's blog
A lot of stuff about the field of arithmetic combinatorics in general (from a course given by Terry Tao)

If you want other examples of the relationship between measure theory and number theory, let me know and I'll post more...
 Sci Advisor P: 2,340 Welcome to PF, Xevarion! And hurrah, it's a a joy to see such a hip newbie I just mentioned the Szemeredi theorem a few months ago in another thread. And of course we want to see more examples! BTW, for those who want to run to their local math library, good for you; some relevant undergraduate textbooks are: G. J. O. Jameson, The Prime Number Theorem, LMS student texts 53, Cambridge University Press, 2003. Mark Pollicott and Michiko Yuri, Ergodic Theory and Dynamical Systems, LMS student texts 42, Cambridge University Press, 1998. Two superb graduate level textbooks which offer discussions of the Szemeredi theorem in the context of graph theory are Bela Bolobas, Modern Graph Theory, GTM 184, Springer, 1998 (one of the best books ever published in any subject!) Reinhard Diestel, Graph Theory, GTM 173, 2nd Ed., Springer, 1999. Note that one could not find two graduate level textbooks with fewer mathematical prerequisites, so I encourage adventurous readers of all backgrounds to take a look if they have the opportunity.
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 Quote by Chris Hillman And of course we want to see more examples!
O.K. :)

Another thing I learned about recently is one of Melvyn Nathanson's pet problems. I think it's very interesting. Here's the idea: if A is a set of integers, define
A + A = {a1 + a2 | a1, a2 in A} (the sum set)
A - A = {a1 - a2 | a1, a2 in A} (the difference set)
Since a1 - a2 is not the same as a2 - a1, but a1+a2 = a2+a1, one would expect that A-A is bigger than A+A. Perhaps surprisingly, one can actually find sets with a larger sum set than difference set. For example, A = {0, 2, 3, 4, 7, 11, 12, 14} has sum set 0-28 missing just 1, 20, and 27; but A-A is -14 to 14 missing +-6, +-13.

So, now that we know these weird sets exist, we might at least hope that there are not many of them. This is where measure theory comes in. How do we define "not many"? We'd like a statement of the following type:
"The measure of the collection of sets A with more sums than differences and with A subset of [1, N] goes to 0 and N goes to infinity."
What measure do we use? Apparently, somebody proved that if you use the uniform measure (which gives the same measure to any subset of [1, N]), actually some positive fraction of sets have more sums than differences! But you can put other measures on those collections of sets (possibly more natural ones, although less simple). And in some of those other measures, there are actually "few" such "bad" sets, as we'd hope.

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Another thing that's worth pointing out is that Furstenberg proved Szemeredi's theorem by a kind of transference principle. He found a purely measure-theoretic statement which implied the number theory result. The amazing thing is that the two theorems are actually equivalent. That is, the number theoretic result also implies the seemingly more general ergodic theorem!

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If I think of anything else worth mentioning, I'll post it here too. Unfortunately I've been spending a lot of time on arithmetic combinatorics lately so that's all that comes to mind right now!
 Sci Advisor P: 2,340 [EDIT: rambling class p rapids "stream of conciousness" largely removed] Many ergodic theorists have been interested in Penrose tilings and the theory of aperiodic tiling spaces generally. One place where this theory intersects at least two of the topics you mentioned (Fourier analysis and difference sets) is in the application of Meyer sets; see R. V. Moody, "Meyer Sets and Their Duals", The Mathematics of Long-Range Aperiodic Order, Kluwer, 1997. In this paper, Moody establishes seven equivalent characterizations of Meyer sets (a generalization of lattice suitable for a kind of harmonic analysis) is that a relatively dense set $\Latex \subset R^n$ is a Meyer set iff the difference set $\Lambda-\Lambda$ is uniformly discrete. Here, $\Lambda$ is relatively dense if there is a compact set such that $R^n = K + \Lambda$. So taking n=1 and K=[0,1], the integers are relatively dense in R. And $\Lambda$ is uniformly discrete if there is an open neighborhood of the origin, U, such that the difference set misses U. I am thinking of the example you gave as a "patch" from a "tiling" with prototiles of length 1,2,4, namely oo*o****oooo*o In the rambling first version of this post, I seized upon translation invariants aspects of your comments and ignored everything else. This prompts me to inquire whether one can arbitrage sum sets versus difference sets? I've always been intrigued by Moody's paper but AFAIK this point of view has not been followed up, and it occurs to me that AFAIK Szemeredi's theorem has not been applied directly to tiling theory. I think that I'm trying to suggest that it might be suggestive to try to interpret additive phenomena in the integers using some of the language of tiling theory, which might suggest some interesting problems. In addition, while IMO the general theory of tiling has not yet appeared, I expect it should provide a scheme for founding mathematics upon tilings rather than upon sets. If so from this POV it would not be surprising that "additive phenomena" can describe seemingly unrelated phenomenon. The discovery of Penrose tilings gave rise to a great deal of interest in how unexpectedly rigid long range order can result from simple local rules. One could say that Szemeredi theorem concerns unexpectedly unavoidable order of a kind. So there does seem to be at least a vague spiritual connection. Incidently, Bombieri, who worked on analytic topics related to some of the topics you mentioned, was one of those who became intrigued by Penrose tilings!