Xll Forewo heart—basic analyti c inequalities , probabilisti c reasoning , an d the connections wit h geometri c combinatoric s an d numbe r theory . We begi n th e boo k b y introducin g th e reade r t o th e Cauch Schwartz an d Holde r inequalities . Instea d o f continuin g o n t o t endless, albei t interesting , worl d o f inequalities, we immediately pu sue applications to geometric problems. W e hope that th e natural a peal an d beaut y o f these connections will help us make the case th far fro m bein g solel y a n exercis e i n symbo l manipulation , Cauch Schwartz an d othe r dr y lookin g estimate s reflec t fundamenta l phys cal realities that ca n be appreciated o n many levels. Fo r example, show tha t th e Cauchy-Schwart z inequalitie s ca n b e use d t o estima the numbe r o f incidences o f points an d lines , and size s of projectio of discrete point sets . I n the cours e of discussing projections, w e q etly sneak i n the notio n of interpolation, which i s so fundamental a unavoidable i n researc h harmoni c analysis . Whe n presente d fo r i own sake , thi s concep t ca n appea r dr y an d specialized . I n th e co text o f a concrete problem, however , i t is instead a t wors t a necessa evil neede d t o resolv e th e proble m a t hand . Thes e idea s an d the variants occup y th e firs t fou r chapter s o f the book . In chapter s 5-8 , w e move o n t o th e finite field setting , explain in detail withou t an y nee d fo r prerequisites , thu s simplifyin g th e c culations an d eliminatin g th e nee d fo r muc h formalism . Thi s allo us to present muc h of what i s known on the Besicovitch-Kakey a co jecture, on e o f th e mos t importan t an d centra l problem s o f mode harmonic analysi s whic h connect s th e siz e o f a se t wit h th e numb of "lin e segments" o f different "slopes " contained within . Chapter s and 1 0 are dedicated t o problem s an d idea s that requir e basic cou ing an d probabilisti c reasoning , whic h w e the n connec t wit h som interesting question s i n the theor y o f numbers , thu s puttin g a diffe ent perspectiv e on calculations an d concept s introduced earlie r in t manuscript. Chapter s 1 1 and 1 2 of the book are dedicated to trigon metric sums , an d sum s an d integral s wit h application s t o problem in geometr y an d numbe r theory . We d o no t ai m fo r th e slickes t proof s o r eve n th e mos t elega presentation. Th e ide a i s to ge t th e reade r t o becom e excite d abo research mathematic s b y observing the process in which ideas evol

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