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A geometric method for the approximation by nearest integer continued fractions. (Eine geometrische Methode bei der Approximation durch Kettenbrüche nach dem nächsten Ganzen.) (German) Zbl 0837.11003

Let \(A_n/B_n\) be the convergents in the nearest integer continued fraction expansion for some real irrational number \(a\), and define \(\lambda_n\) by \(|a - A_n/B_n |= 1/(\lambda_n B^2_n)\). H. Jager and C. Kraaikamp [Indag. Math. 51, 289-307 (1989; Zbl 0695.10029)] developed a geometric method to show that \(\max (\lambda_{n - 1}, \lambda_n, \lambda_{n + 1}) > (11 + 5 \sqrt 5)/10\), for any \(n\). The lower bound falls short of Borel’s bound \(\sqrt 5\) in an analogous estimate for simple continued fraction approximations.
The authors use the method to show that the lower bound is best possible, and that Borel’s bound cannot be achieved by increasing the number of consecutive convergents. This has also been proved by J. Tong [Math. Scand. 71, 161-166 (1992; Zbl 0787.11028)] by another method.

MSC:

11A55 Continued fractions
11J70 Continued fractions and generalizations
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