By Terence Tao
Conventional Fourier research, which has been remarkably potent in lots of contexts, makes use of linear section capabilities to review services. a few questions, comparable to difficulties regarding mathematics progressions, certainly result in using quadratic or better order levels. better order Fourier research is a topic that has develop into very energetic just recently. Gowers, in groundbreaking paintings, built a number of the easy recommendations of this conception as a way to supply a brand new, quantitative facts of Szemerédi's theorem on mathematics progressions. even if, there also are precursors to this concept in Weyl's classical thought of equidistribution, in addition to in Furstenberg's structural concept of dynamical structures. This booklet, that's the 1st monograph during this sector, goals to hide all of those subject matters in a unified demeanour, in addition to to survey probably the most contemporary advancements, reminiscent of the appliance of the idea to count number linear styles in primes. The e-book serves as an creation to the sector, giving the start graduate pupil within the topic a high-level evaluation of the sector. The textual content specializes in the best illustrative examples of key effects, serving as a significant other to the prevailing literature at the topic. there are lots of workouts with which to check one's wisdom.
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