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By Liouville's theorem on Diophantine approximation, (https://en.wikipedia.org/wiki/Diophantine_approximation), there exists a constant $c$ such that $$|\sqrt{2}-{m_1\over m_2}|>{c\over m_2^2}.$$ for every choice of integers $m_1,m_2$ such that $m_2>0$ Consequently, for every choice of integers such that $m_2>0$: $$|m_2\sqrt{2}-m_1|>{c\over ...


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from Khinchin The integers $a_k$ are called the elements, with the possible exception of $a_0$ they are positive. Here we go, on page 7 we find his numbering, $$ q_{k+1} = a_{k+1} q_k + q_{k-1}. $$ On page 9 we have Theorem 9, $$ \left| \alpha - \frac{p_k}{q_k} \right| < \frac{1}{q_k q_{k+1}} < \frac{1}{ a_{k+1} q_k^2} $$ If the ...


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Lets begin by noticing that we only have to concern ourselves with $r\in [0, 1]$: any number that can be so approximated will differ by an an integer from a number in that interval. Now, consider the sets $$A_q = \left[0, \frac{C}{q^3} \right) \cup \left( 1 - \frac{C}{q^3}, 1 \right]\cup \bigcup_{i=1}^{q-1} \left( \frac{i}{q} - \frac{C}{q^3}, \frac{i}{q} + ...



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