# Measure theory and number theory?

1. Nov 15, 2007

### pivoxa15

How is measure theory associated with number theory, if at all.

If they are connected, can anyone give a link?

2. Nov 15, 2007

### mathman

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.

3. Nov 16, 2007

### mathman

Post script:

Number theory was studied by the ancient Greeks (Euclid et al). Measure theory development is around 100 years old.

4. Nov 16, 2007

### pivoxa15

There is analytical number theory. Analysis is related to measure theory and so is integration so there is an indirect link at least.

5. Nov 16, 2007

### Kummer

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.

6. Nov 17, 2007

### pivoxa15

I assume you are referring to ring theory in algebra.

7. Nov 17, 2007

### quasar987

8. Nov 17, 2007

### 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.

9. Nov 17, 2007

### 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.

10. Nov 19, 2007

### mathwonk

read andre weils book, basic number theory, where you will see haar measure treated as one of the first topics (page 3).

(in fact this appears in the references for quasar's link)

or see fourier analysis on number fields, by ramakrishnan.

thus to say the two subjects have nothing to do with one another is to me incorrect.

John Tate
• Ten things you may not know about Wikipedia •
For other people with similar names, see John Tate (disambiguation).
John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. He wrote a Ph.D. at Princeton in 1950 as a student of Emil Artin, was at Harvard University 1954-1990, and is now at the University of Texas at Austin.
Tate's thesis, on the analytic properties of the class of L-functions introduced by Erich Hecke, is one of the relatively few such dissertations that have become a by-word. In it the methods, novel for that time, of Fourier analysis on groups of adeles, were worked out to recover Hecke's results.
Subsequently Tate worked with Emil Artin to give a treatment of class field theory based on cohomology of groups, explaining the content as the Galois cohomology of idele classes, and introduced Tate cohomology groups. In the following decades Tate extended the reach of Galois cohomology: Poitou-Tate duality, abelian varieties, the Tate-Shafarevich group, and relations with algebraic K-theory.
He made a number of individual and important contributions to p-adic theory: the Lubin-Tate local theory of complex multiplication of formal groups; rigid analytic spaces; the 'Tate curve' parametrisation for certain p-adic elliptic curves; p-divisible (Tate-Barsotti) groups. Many of his results were not immediately published and were written up by Jean-Pierre Serre. They collaborated on a major published paper on good reduction of abelian varieties.
The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture. They relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tate-twisted action) that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.
Tate has had a profound influence on the development of number theory through his role as a PhD advisor. His students include Joe Buhler, Benedict Gross, Robert Kottwitz, James Milne, Carl Pomerance, Ken Ribet and Joseph H. Silverman.
He was awarded a Wolf Prize in Mathematics in 2002/3.

Sato-Tate conjecture
Sato-Tate measure
Tate module
Néron–Tate height
Selected publications

J. Tate, Fourier analysis in number fields and Hecke's zeta functions (Tate's 1950 thesis), reprinted in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2

Last edited: Nov 19, 2007
11. Nov 19, 2007

### Office_Shredder

Staff Emeritus
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

12. Nov 20, 2007

### mathwonk

13. Nov 21, 2007

### Xevarion

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...

14. Nov 21, 2007

### Chris Hillman

Everyone Should Know a Something about Ergodic Ramsey Theory

Welcome to PF, Xevarion!

And hurrah, it's a a joy to see such a hip newbie I just [post=1416125]mentioned[/post] 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.

Last edited: Nov 21, 2007
15. Nov 21, 2007

### Xevarion

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.

----

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!

----

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!

16. Nov 21, 2007

### Chris Hillman

A Tiling Interpretation?

[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!

Last edited: Nov 21, 2007
17. Nov 21, 2007

### Xevarion

Unfortunately I don't actually know that much about tiling, so I am not sure exactly what kind of application you want to make here. I can at least tell you a few relevant things:
1) There is a generalization of Szemeredi's theorem to Z^n. However as far as I know the only proof of this right now is ergodic in nature, and therefore there are no bounds.
2) People also do try to prove results of this nature about thin subsets of the reals. So if you're actually interested in difference sets of some collection of vectors in R^n, there should be some theory too. But I don't think it is as rich or well-studied as in the cases of Z, Z/nZ, or finite fields. In particular, I don't know much about what's known in R^n for n>1.
3) Somehow from a measure-theoretic perspective, R^n (n > 1) suddenly gets complicated. They contain too much stuff. You can have integer-dimensional sets that are nice, but don't look that much like copies of R^k. Possibly if you had some set and you wanted to impose some additive structure on it, that would force it to be contained in some proper (affine) subspace... I dunno :-o

edit: by the way, you might be interested in reading about some of the recent results of Bourgain, Gamburd, and Sarnak about lattices and orbits. E.g. this article (sorry I don't have a better link)

Last edited: Nov 21, 2007
18. Nov 21, 2007

### Chris Hillman

Neither am I (that's why I wrote "mumble, mumble"; I was thinking aloud).

I think that is worth elaborating on (better you than me!).

Regarding expanders, I have "read" Davidov, Sarnak, Valette, Elementary Number Theory, Group Theory, and Ramanujan Graphs but this wasn't as enlightening as I had hoped... I sent you a PM (private message=internal "email" at PF=Physics Forums).

Last edited: Nov 21, 2007
19. Nov 21, 2007

### morphism

Really? I find that hard to believe after reading his book on functional analysis.