Recurrence in Ergodic Theory and Combinatorial Number Theory
Topological dynamics and ergodic theory usually have been treated independently. H. Furstenberg, instead, develops the common ground between them by applying the modern theory of dynamical systems to combinatories and number theory.Originally published in 1981.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Combinatorial Number Theory and Additive Group Theory
Additive Group Theory and Non-unique Factorizations.- Notation.- Basic concepts of non-unique factorizations.- The Davenport constant and first precise arithmetical results.- The structure of sets of lengths.- Addition theorems and direct zero-sum problems.- Inverse zero-sum problems and arithmetical consequences.- Sumsets and Structure.- Notation.- Cardinality inequalities.- Structure of sets with few sums.- Location and sumsets.- Density.- Measure and topology.- Exercises.- Thematic seminars.- A survey on additive and multiplicative decompositions of sumsets and of shifted sets.- On the detailed structure of sets with small additive property.- The isoperimetric method.- Additive structure of difference sets.- The polynomial method in additive combinatorics.- Problems in additive number theory, III.- Incidences and the spectra of graphs.- Multi-dimensional inverse additive problems.
Ultrafilters and combinatorial number theory
Our concern is with two areas of mathematics and a, possibly surprising, intimate connection between them. One is the branch of combinatorial number theory which deals with the ability, given a finite partition of ℕ, to find sums or products of certain descriptions lying in one cell of that partition. The other is the branch of set theoretic topology dealing with the existence of ultrafilters on ℕ which have specified properties. We shall present and, to the extent feasible, prove those major results in the former area with which we are familiar and many of the related results in the latter area.
Some Problems and Results on Combinatorial Number Theory
I have written many papers with similar titles during my long life. I will try to write this paper in such a way that it will not entirely be contained in the union of the set of my previous papers and that a t least some of the open problems I state will not be entirely hopeless. Perhaps the most interesting and significant results are those connected with van der Waerden’s and SzemerCdi’s theorem, but since I and others have written a great deal about these questions, I will include only a short discussion of these problems at the end of the paper. For a rich source of solved and unsolved problems in combinatorial number theory, see [ 10,201. First of all I mention a few old problems and results. First a very simple old well-known result of mine.
Zero-sum sets of prescribed size
Erdős, Ginzburg and Ziv proved that any sequence of 2n−1 integers contains a subsequence of cardinality n the sum of whose elements is divisible by n. We present several proofs of this result, illustrating various combinatorial and algebraic tools that have numerous other applications in Combinatorial Number Theory. Our main new results deal with an analogous multi dimensional question. We show that any sequence of 6n − 5 elements of Zn ⊕ Zn contains an n-subset the sum of whose elements is the zero vector and consider briefly the higher dimensional case as well.
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