By Terence Tao
Conventional Fourier research, which has been remarkably potent in lots of contexts, makes use of linear section services to check features. a few questions, akin to difficulties related to mathematics progressions, obviously bring about using quadratic or larger order stages. better order Fourier research is a topic that has turn into very energetic only in the near past. Gowers, in groundbreaking paintings, built the various simple ideas of this thought with a view to provide a brand new, quantitative evidence of Szemerédi's theorem on mathematics progressions. notwithstanding, there also are precursors to this idea in Weyl's classical thought of equidistribution, in addition to in Furstenberg's structural idea of dynamical structures. This publication, that's the 1st monograph during this region, goals to hide all of those themes in a unified demeanour, in addition to to survey one of the most fresh advancements, resembling the applying of the speculation to count number linear styles in primes. The publication serves as an advent to the sector, giving the start graduate pupil within the topic a high-level assessment of the sphere. The textual content makes a speciality of the easiest 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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