KVPY 2016 SB Question 2

Algebra Level 4

For any real number $r$ , let $A_r = \{ e^{i \pi r n} : n \text{ is a positive integer} \}$ be a set of complex numbers.

Which of the following is true?

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A_1

is a finite set and

A_{\frac1\pi} , A_{0.3}

are infinite sets

A_1, A_{0.3}

are finite sets and

A_{\frac1\pi}

is an infinite set

A_1, A_{\frac1\pi}, A_{0.3}

are all finite sets

A_1, A_{\frac1\pi}, A_{0.3}

are all infinite sets

1 solution

Guillermo Templado
Feb 25, 2017

relevant wiki: DeMoivre's theorem $A_1 = \{-1, 1\}$ $A_{0.3} = \{e^{i \pi 0.3 n}; \space n \in \mathbb{N} \text { and } 1 \leq n \leq 20\} = \{-1, 1, e^{i \pi \cdot 0.3}, e^{i \pi \cdot 0.6},...\}$ has 20 elements , and $\large A_{\frac{1}{\pi}}$ is an infinite set because $f : (\mathbb{R}, +) \to (\mathbb{T}, \cdot)$ is a surjective homomorphism of groups with kernel $2\pi \mathbb{Z}$ ,where $f(t) = e^{it}$ and $\mathbb{T}$ is the unitary circumference (radius 1) in $\mathbb{C}$ .

This means if $m,n \in \mathbb{N}$ with $m \neq n$ satisfied $e^{im} = e^{in}$ then there would exist $c \in \mathbb{Z}$ such that $e^{i(m - n)} = 1 = e^{i 2\pi c} \Rightarrow m = n + 2\pi c$ (Impossible = No = Contradiction)

Bonus.- Why $A_{0.3}$ has 20 elements?

KVPY 2016 SB Question 2

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