# Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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### Badly Approximable Numbers: Lower Bound for Sum of Reciprocals of Fractional Parts

I am having a bit of trouble proving the following fact that I have seen in a couple of references on Diophantine approximation. Let $||x|| = \min\{|x - p| ~|~ p \in \mathbb{Z}\}$. Recall that a ...
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### Understanding the Proof of Chapter 2, Theorem 8 of Lang's Book on Diophantine Approximations

I am having a bit of trouble understanding the inductive summation step in the proof of Theorem 8 of Chapter 2 in Lang's Introduction to Diophantine Approximations (p. 31). In particular, Lang shows ...
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### For $\{n\pmod{2 \pi}: n \in \{\ 0, 1, 2 ... 2^x\ \} \}$, is it possible to put a bound in terms of $x$ on the smallest difference between elements?

Is it possible to put some sort of bounds on the smallest difference between elements for a limited range of integers $\mod 2\pi$? $$A =\bigl\{n\pmod{2\pi} : n \in \{\ 0, 1, 2, ... 2^x\ \} \bigl\}$$ ...
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