Notation The harmoni c analysi s pursue d her e involve s th e rea l lin e K, th e circl e grou p T = R/Z , o r the finite Fourier transform (calle d "additiv e characters" b y number theorists). W e le t e(x) = e 2nlx denote th e comple x exponentia l wit h perio d 1 , s o tha t i f / £ L 1 (T) the n it s Fourier coefficient s ar e give n b y th e formul a /(* )= [ f(x)e(-kx)dx. Similarly, th e Fourie r coefficient s o f a Bore l measur e / i on T w e take t o b e fi(k) = / e(—kx)d/j,(x). We let [x] and {x} denot e th e integra l par t o f x an d th e fractiona l par t o f x, respectively. Thu s x = [x] + {x} wit h [x] Z an d 0 {x} 1 . I n addition , w e let \\x\\ denote th e distanc e fro m x t o th e neares t integer , ||x| | = min nG z \x n\. Thus ||x| | i s the natura l nor m o n T . The relatio n / ^ C g mean s exactl y th e sam e thin g a s / = 0(g) tha t is , there i s a n absolut e constan t C suc h tha t |/ | Cg fo r al l value s o f th e fre e variables unde r consideration . I f th e implici t constan t C i s allowe d t o depen d on a parameter k, the n suc h dependence ma y b e indicated b y writing / Cf c g or / = O k (g). More specialize d notation , appropriat e t o variou s topics , i s develope d i n in - dividual chapters . Suc h notatio n shoul d no t b e expecte d t o b e consisten t fro m one chapte r t o another . Fo r example , 6 in Chapte r 1 is quite differen t fro m 6 in Chapter 2 . Xlll
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