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In the second lecture (based on Gowers’s joint work with Julia Wolf) we were introduced to decomposition theorems. A decomposition theorem for the norm can be stated as follows: if is a function (on either or ) with , there is a decomposition , where are “generalized quadratic phase functions” and are error terms with and small. This can be deduced from the inverse theorem of Green-Tao; in fact a similar statement was already implicit in their work, based on the energy increment argument. Tim presented a different approach to deducing decomposition theorems from inverse theorems, based on functional-analytic arguments involving the geometry of normed spaces (specifically, a variant of the Hahn-Banach theorem).

This can be applied to the question of counting solutions to systems of linear equations in sets. Let’s say that we are interested in finding sensible conditions under which a set will have the “statistically correct” number of solutions to a system of linear equations. For instance, if it is 4-term arithmetic progressions that we are concerned with, then uniformity is sufficient (and, in general, necessary). Green and Tao prove a more general result of this type: they define the complexity of a system of linear forms, and prove that systems of complexity are controlled by norms.

Gowers and Wolf, however, do not stop there. Suppose that, instead of 4-term progressions, we are interested in configurations of the form, say, . The complexity of this system in the sense of Green-Tao is 2, hence a set uniform in the norm will contain the “right” number of such configurations. Gowers and Wolf, however, can prove that uniformity already guarantees the same conclusion! The difference between the two examples? The squares are linearly dependent, whereas are not. Gowers and Wolf prove that such “square independence” is in fact both sufficient and necessary for a system of complexity 2 to be controlled by the $U^2$ norm. The proof is based on the decomposition theorem described earlier.

Topics in Harmonic Analysis and Ergodic Theory Joseph M. Rosenblatt, Alexander M. Stokolos, Ahmed I. Zayed ISBN: 0821842358 Paperback American Mathematical Society Usually despatched within 3 to 9 days

There are strong connections between harmonic analysis and ergodic theory. A recent example of this interaction is the proof of the spectacular result by Terence Tao and Ben Green that the set of prime numbers contains arbitrarily long arithmetic progressions. The breakthrough achieved by Tao and Green is attributed to applications of techniques from ergodic theory and harmonic analysis to problems in number theory.Articles in the present volume are based on talks delivered by plenary speakers at a conference on Harmonic Analysis and Ergodic Theory (DePaul University, Chicago, December 2-4, 2005). Of ten articles, four are devoted to ergodic theory and six to harmonic analysis, although some may fall in either category. The articles are grouped in two parts arranged by topics. Among the topics are ergodic averages, central limit theorems for random walks, Borel foliations, ergodic theory and low pass filters, data fitting using smooth surfaces, Nehari's theorem for a polydisk, uniqueness theorems for multi-dimensional trigonometric series, and Bellman and s-functions.In addition to articles on current research topics in harmonic analysis and ergodic theory, this book contains survey articles on convergence problems in ergodic theory and uniqueness problems on multi-dimensional trigonometric series.