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By Klin M., et al.

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2 Farey Table 51 a a a+a a a We call a+a b+b the mediant of b and b , and write b+b =: b ⊕ b . The nth row is called the nth Farey row, and the collection of rows the Farey table. The Farey table, another old friend from elementary number theory, exhibits some remarkable properties: A. Every row is strictly increasing from 01 to 11 , and consists of rational p q−p numbers between 0 and 1. The numbers appear in symmetric pairs q , q around the middle. 8) in the next row. But this is immediate, since a a a+a a ⇐⇒ ab + ab < ab + a b ⇐⇒ ab < a b ⇐⇒ , < < b b+b b b and similarly for the right-hand inequality.

With ai ∈ {1, 2} for all i is badly approximable and hence has order of approximation precisely 2. In particular, every α ∈ Q with L(α) < 3 is badly approximable. We will see later that every irrational α with L(α) < 3 is, in fact, equivalent to a quadratic irrational. So this result is again a special case of Liouville’s theorem. Whether there exist algebraic numbers of degree ≥ 3 that are badly approximable is an open problem. It is common belief that the answer is no. Notes 29 Notes Continued fractions and Diophantine approximations are treated in many number theory books.

An−1 , αn ] > [a1 , . . , an−1 , βn ] (−1)n−1 αn > (−1)n−1 βn (−1)n αn < (−1)n βn (−1)n an < (−1)n bn , since αn < βn if and only if an < bn holds. 25. Two irrational numbers α and β are called equivalent, denoted by α ∼ β, if their continued fraction expansions eventually coincide, that is, α = [a0 , a1 , . . , ak , γ], β = [b0 , b1 , . . , b , γ]. Clearly, ∼ is an equivalence relation. 26. Equivalent numbers α and β have the same Lagrange number, L(α) = L(β). 4 Lagrange Spectrum and Continued Fractions 25 Proof.

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Algebraic combinatorics in mathematical chemistry by Klin M., et al.

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