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).
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