Keith Briggs

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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 (2003), 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. This website uses no cookies. This page was last modified 2024-01-21 10:57 by .