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By Stein M.R., Dennis R.K. (eds.)

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Extra resources for Algebraic K-Theory and Algebraic Number Theory

Example text

To illustrate this, consider the periodic continued fraction 1ր + 1ր + 1ր + 1ր 2 3 1 4 The associated periodic strip in the Farey diagram is the following: We would like to compute the element T of LF (Z) that gives the rightward translation of this strip that exhibits the periodicity. A first guess is the T with matrix 4 19 9 43 since this sends 1/0, 0/1 to 4/9, 19/43 . This is actually the correct T since it sends the vertex 1/1 just to the right of 1/0 , which is the mediant of 1/0 and 0/1 , to the vertex (4 + 19)/(9 + 43) just to the right of 4/9 , which is the mediant of 4/9 and 19/43 .

Recall the original example that we did: 67 − − − 24 19 = 2+ − − − − = 2+ 24 1 − −−− 24 / = 2+ 1 − − − − − −−− 1 + 5/ = 2+ 19 19 1 −− − − − −−1− 1 + − − − − − − − − 19 / 5 = 2+ 1 −− − − − −− − − − −− 1 1 + − − − − − − − − 3+ 4 /5 = 2+ 1 −− − − − −− − − − −− 1 1 + − −−−−−− − 1 3 + − − − − − 5/ 4 = 2+ 1 − − − − − − − − − − − − − − − − − − − − − − − − 1 1 + − − − − − − − − − − − − − − − − − 1 3 +− − − − − − − − − − 1 1 + − − − 4 The sequence of steps is the following: (1) Write x = a0 + r1 where a0 is an integer and 0 ≤ r1 < 1 (2) Write 1/r1 = a1 + r2 where a1 is an integer and 0 ≤ r2 < 1 (3) Write 1/r2 = a2 + r3 where a2 is an integer and 0 ≤ r3 < 1 and so on, repeatedly.

We can get good rational approximations to 2 by computing the convergents in its continued fraction 1 + 1ր2 + 1ր2 + 1ր2 + · · · . It’s a little easier to compute the √ convergents in 2 + 1ր2 + 1ր2 + 1ր2 + · · · = 1 + 2 and then subtract 1 from each of these. For 2 + 1ր2 + 1ր2 + 1ր2 + · · · there is a nice pattern to the convergents: 2 5 12 29 70 169 408 985 , , , , , , , , ··· 1 2 5 12 29 70 169 408 Notice that the sequence of numbers 1, 2, 5, 12, 29, 70, 169, · · · is constructed in a way somewhat analogous to the Fibonacci sequence, except that each number is twice the preceding number plus the number before that.

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