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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2005 • Question 1

Question

Math 6210 • Real Analysis 2005 • Question 1

Consider a measurable subset \(A\) of the circle \(\{z\in \mathbb C\mid |z|=1\}\). Fix a real number \(\theta \) so that \(\theta /\pi \) is irrational. Define \[ A_\theta =\{e^{i\theta }z\mid z\in A\}, \] the rotation of \(A\) through \(\theta \) radians. Prove that if \[ A=A_\theta \] then either \(A\) has measure zero or else \(A\) is the complement of a set of measure zero. (Hint: consider the Fourier series of \(\chi _A-\chi _{A_\theta }\); here \(\chi _S\) denotes the characteristic function of \(S\).)

Raw LaTeX

Consider a measurable subset \(A\) of the circle
\(\{z\in \mathbb C\mid |z|=1\}\). Fix a real number \(\theta\) so that
\(\theta/\pi\) is irrational. Define
\[
A_\theta=\{e^{i\theta}z\mid z\in A\},
\]
the rotation of \(A\) through \(\theta\) radians. Prove that if
\[
A=A_\theta
\]
then either \(A\) has measure zero or else \(A\) is the complement of a set of
measure zero. (Hint: consider the Fourier series of
\(\chi_A-\chi_{A_\theta}\); here \(\chi_S\) denotes the characteristic function of
\(S\).)

Topic Tags

Automorphism Groups and Biholomorphisms Fourier Analysis and Convolution Measure Theory Zeros of Holomorphic Functions and the Identity Theorem