Algebraic Numbers and Harmonic Analysis
Diophantine approximations to real numbers. Some classical results in diophantine approximations. Measure-teoretical methods in diophantine approximations. Diophantine approximations and additive problems in locally compact abelian groups. Uniqueness of representation by trigonometric series. Problems on a-periodic trigonometric sums. Special trigonometric series (complex methods). Special trigonometric series (group-theoretic methods). Pisot numbers and spectral synthesis. Ultra-thin symmetric sets.
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Chapter I Diophantine approximations to real numbers
Chapter II Diophantine approximations and additive problems in locally compact abelian groups
Chapter III Uniqueness of representation by trigonometric series
Chapter IV Problems on aperiodic trigonometric sums
Chapter V Special trigonometric series complex methods
Chapter VI Special trigonometric series grouptheoretic methods
Chapter VII Pisot numbers and spectral synthesis
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algebraic integer algebraic number ﬁeld Assume Banach algebra bounded continuous function Chapter coefﬁcients coherent set compact set compact subset complex numbers complex valued bounded complex valued continuous constant contained continuous function convergence deﬁned deﬁnition discrete dual group element equivalent exists ﬁnd ﬁnite set ﬁnite subset ﬁnite sums ﬁrst ﬁxed following lemma Fourier series Fourier transform frequencies belong function f function whose spectrum Haar measure harmonious sets Hence homomorphism implies increasing sequence l.c.a. group Laplace transform Lebesgue measure Lemma Let f Let G mean periodic function metric modulo norm p-adic Pisot number positive real number proof of Theorem proved Radon measure rational integers relatively dense Salem number satisﬁes Section set of frequencies set of real spectra lie spectrum lies subset of G sufﬁces sufﬁciently large sums whose frequencies tends to inﬁnity Theorem XI tion topological Sidon set topology trigonometric sum uniformly weak-star weak-star topology