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The uploaded image contains a mathematical exercise involving complex numbers, particularly focusing on properties of the unit circle \( U \), arguments of complex numbers, and the discrete set of \( n \)-th roots of unity.

Here is an outline of the tasks in the problem:

1. **Basic Definitions**:
   - \( U \) represents the set of complex numbers of modulus 1 (the unit circle).
   - \( U_n \) represents the set of the \( n \)-th roots of unity.
   - The notation \( d(z, z') \) defines the distance between two complex numbers \( z \) and \( z' \), presumably on the unit circle.
   - \( \arg(z) \) and \( \text{Arg}(z) \) are defined for the argument of a complex number, with \( \text{Arg}(z) \) specifically the principal argument (i.e., in \( [0, 2\pi) \)).

2. **The Main Problem: Set \( V \)**
   - The set \( V = \{z_n = e^{i\theta n} / n \in \mathbb{Z}\} \) is introduced.
   - The goal is to study this set \( V \) and its properties.

3. **Tasks**:
   - **1.** Compute the distance \( d(e^{i\alpha}, e^{i\beta}) \) and express it in terms of \( \alpha \) and \( \beta \).
   - **2.** Define a set \( A = \{n \in \mathbb{N} / z_n = 1\} \) and show that it has a smallest element.
   - **3.** Show that \( z_s \) and \( z_t \) are distinct when \( \theta / \pi \) is irrational.
   - **4.** Establish results about approximations of arguments and distances within \( U \).

This appears to be a rigorous and detailed exploration of arguments and distances on the unit circle, using properties of complex exponentials, the roots of unity, and potentially number theory elements like rational and irrational numbers.

Would you like to focus on a specific part of this problem, or would you like further explanation or a breakdown of the solutions to each question? 

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Here are 5 related questions to deepen understanding:
1. How is the distance between two points on the unit circle related to their angular separation?
2. What properties of roots of unity can be used to show that \( A \) has a smallest element?
3. Why is the irrationality of \( \theta / \pi \) crucial for proving that the elements \( z_s \) and \( z_t \) are distinct?
4. How does partitioning the unit circle help in estimating arguments and distances between complex numbers?
5. How can inequalities involving trigonometric functions be used to control distances on the unit circle?

**Tip**: When working with arguments of complex numbers, always keep in mind that they are only defined modulo \( 2\pi \), meaning that angles differing by multiples of \( 2\pi \) represent the same complex number.

The problem involves complex numbers on the unit circle, focusing on the arguments, roots of unity, and distances between complex numbers. The user is tasked with proving certain properties of these numbers, such as distinctness of roots, existence of certain values, and partitioning of the unit circle.

Explore a complex number problem involving roots of unity, arguments, and distances on the unit circle. This problem focuses on distinctness of roots, proving properties of arguments, and partitioning the unit circle. Suitable for advanced undergraduate or graduate students.

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