Backward fractal signature of irrationals
franktaw at netscape.net
franktaw at netscape.net
Thu Oct 18 18:13:43 CEST 2007
First, you need to realize that, while all signature sequences are
fractal sequences, the reverse is not true. There are many fractal
sequences which are not signature sequences. In fact, I have
conjectured that, essentially, a sequence is a signature sequence
iff it is both fractal and also equal to its lower-trimmed subsequence.
(The "essentially" is to cover the cases where the sequence is the
equivalent of a signature sequence for a positive rational number.
There are two such sequences for each positive rational, one with
equal values of (c + d theta) sorted in increasing order of c, and
one in increasing order of d.)
For those sequences which are signature sequences, if you look
at the first index where they differ, the one that is larger there
corresponds to a larger value for theta. So you can determine
what theta is by performing a binary search. This is most efficient
using Farey sequences. That is, having determined that your
number is between a/b and c/d, check next to see if it is above
or below (a+c)/(b+d).
I believe that if you take the positions of the 1's in the signature
sequence for a number, take their first differences, and subtract
1 from each, you will get the Beatty sequence (see
http://mathworld.wolfram.com/BeattySequence.html ) for the
number. This may provide a slightly faster way to compute the
number from the sequence.
However, note that generally a large number of terms of the
signature sequence are required in order to get even a
moderately accurate estimate of the number. This is especially
true when the continued fraction for the number contains large
values.
Franklin T. Adams-Watters
-----Original Message-----
From: Eric Angelini <Eric.Angelini at kntv.be>
Hello SeqFans -- and sorry again if this is old hat,
another backward-task.
We know what a signature seq is:
http://mathworld.wolfram.com/SignatureSequence.html
Question:
Is it possible, having a fractal seq, to "rewind it"
in order to find from which irrational it comes?
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