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hmwk2 - Math 220A Complex Analysis Solutions to Homework#2...

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Unformatted text preview: Math 220A Complex Analysis Solutions to Homework #2 Prof: Lei Ni TA: Kevin McGown Conway, Page 14, Problem 11. Parts of what follows are adapted from the text Modular Functions and Dirichlet Series in Number Theory by Tom Apostol. There are shorter proofs, but I am trying to avoid compactness arguments. Before giving the solution, we establish some notation. We will write b x c to denote the floor of a real number x ; this is defined to be the greatest integer not exceeding x . We will write { x } to denote the fractional part of x ; this is defined to be x- b x c . In order to solve the problem, we will prove the following: Theorem 1 The set exp(2 πinθ ) | n ∈ Z + is dense in the unit circle if and only if θ is irrational. Taking arguments and normalizing by 2 π , the above theorem is clearly equiva- lent to the following. Theorem 2 The sequence { θ } , { 2 θ } , { 3 θ } ,... is dense in [0 , 1] if and only if θ is irrational. Proof. If θ is rational, then it is easy to see that {{ nθ } | n ∈ Z + } is a finite set and hence it cannot be dense in the unit circle. (A finite subset of C is closed and thus cannot be dense in an infinite set.) Suppose θ is irrational. Without loss of generality, we will assume that θ ∈ [0 , 1]. Indeed, one may replace θ by { θ } after making the observation that { nθ } = { n { θ }} ....
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