By Doug Hensley

This publication areas designated emphasis on persevered fraction Cantor units and the Hausdorff measurement, algorithms and research of algorithms, and multi-dimensional algorithms for simultaneous diophantine approximation. huge computer-generated pics are provided, and the underlying algorithms are mentioned and made on hand.

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Example text

Su[Su] concerning the distribution of {na : 1 < n < q^) mod 1, where (pk/qk) are the successive convergents to an irrational a. T h e result is simplest in t h e case where the fractional p a r t of a is less t h a n 1/2, so we make t h a t assumption. 4, the successive minima of \\qa\\, q>l occur at q = qj,j > 0 and are given by \\qja\\ = \pj\. Su's result begins with the notion of dividing the open unit interval (0,1) into countably many half-open intervals, or bins Bj, j > 0, with Bj : = [\pj\, \pj-i\).

J, then 2J2n=i{na} ~ N = ^(l°S-^)> a result which also follows from the bounds on the discrepancy of the sequence. We close with some illustrations of these sums. There are strong hints of a fractal structure to these sums. r Wf^NiW' •>nm Fig. 2 Fig. 3 Partial sums of {ne} to 4000. Partial sums of {n\/2} to 34000. Continued 48 Fig. 4 Fractions Running total of parity of {fc\/2 - 1/2}. 1 Ergodic Maps Ergodic theory is based on the intuitive idea that a process which shuffles points around may well cause most points to have an orbit that visits every neighborhood of the same size equally often.

One number fell into bin 3, three into bin 2, five into bin 1, and three into bin 0. 0,1,3,5,3) 18 Continued Fractions We form a similar list for each qj and arrive at a table of which the preceding calculation forms the basis for the entries in the fourth row. a j 1 000000 0 0 0 0 0 1 000000 0 0 0 0 1 1 2 000000 0 0 0 1 3 1 5 000000 0 0 1 3 5 3 12 000000 0 1 3 5 15 5 29 000000 1 3 5 15 33 13 70 0 0 0 0 0 1 3 5 15 33 83 29 169 0 0 0 0 1 3 5 15 33 83 197 71 408 0 0 0 1 3 5 15 33 83 197 479 169 985 Inspection reveals an apparent recurrence relation and Su proves it.