000			01905nam a2200229 4500
001			17414662
010			_a 2012023442
020			_a9781470425883
082	0	0	_a515.2433 _bTAO/H
100	1		_aTao, Terence
245	1	0	_aHigher order Fourier analysis
260			_aRhode Island _bAmerican mathematical society _c2012
300			_ax, 187 p.
490	0		_aGraduate studies in mathematics ;
520			_aTraditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in groundbreaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemerédi’s theorem on arithmetic progressions. However, there are also precursors to this theory in Weyl’s classical theory of equidistribution, as well as in Furstenberg’s structural theory of dynamical systems. The book serves as an introduction to the field, giving the beginning graduate student in the subject a high-level overview of the field. The text focuses on the simplest illustrative examples of key results, serving as a companion to the existing literature on the subject. There are numerous exercises with which to test one’s knowledge.
650		0	_aFourier analysis
650		7	_aNumber theory -- Sequences and sets -- Arithmetic combinatorics; higher degree uniformity.
650		7	_aDynamical systems and ergodic theory -- Ergodic theory -- Relations with number theory and harmonic analysis.
650		7	_aNumber theory -- Connections with logic -- Ultraproducts.
650		7	_aNumber theory -- Exponential sums and character sums -- Estimates on exponential sums.
942			_cBK
999			_c67014 _d67014