Keith Briggs

This page was last modified 2013-05-12  

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The worst approximable pair

The problem here is to construct an explicit pair of irrationals with sup-norm simultaneous approximation constant as large as possible, in fact close to 2/7. Very loosely speaking, this is something like a 2-dimensional analog to the golden ratio. The method is described in my preprint and paper (Journal of Number Theory, 103, 71). On 2009-01-27 I obtained an improvement of the largest known approximation constant to 0.285710526941. The computation essentially requires the continued fraction of 2*cos(2*pi/7). I computed 107 partial quotients in about 2 weeks of computing time (using all-integer arithmetic - there is no roundoff error here). The testing of candidate cases uses floating-point arithmetic, for which up to 2*107 mantissa bits were required. These are the successive records (the first seven are in my paper, the last is the new one):

60      [60,1,1,50]   0.285187764997 prec= 196
2927    [22,1,1,22]   0.285315386626 prec= 9906
3629    [272,1,1,215] 0.285572589247 prec= 12329
33880   [81,1,1,78]   0.285626114626 prec= 115757
215987  [124,1,1,129] 0.285667779356 prec= 738768
957740  [460,1,1,415] 0.285680456945 prec= 3279227
1650050 [648,1,1,666] 0.285708196890 prec= 5651092
6034931 [199,1,1,199] 0.285710526941 prec= 20663086
The entries are: index in the continued fraction, four partial quotients starting at that index, approximation constant of the corresponding irrational pair, and mantissa precision in bits used to compute this approximation constant.
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