Backward fractal signature of irrationals

[Max Alekseyev]

On 10/18/07, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>
> 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?

First, you need to define what is "having a fractal sequence".
Elements of such sequence are irrational numbers that in general
cannot be represented precisely in the computer memory.

On the other hand, if the elements are given approximately (e.g., as
floating point numbers) then you can try to determine the irrational
number but, again, only approximately, and in this case there is no
guarantee that this number is uniquely defined.
To do so, take any two elements of the sequence x,y and solve the
integer relation:
m*x + n*y + k = 0
with respect to integers m, n, k.
See http://mathworld.wolfram.com/IntegerRelation.html for details.
Then you can assume that x=n*r + u and y=-m*r + v where r is the
irrational number and integers u,v can be found from the equation m*u
+ n*v + k = 0. This equation is easy to solve if gcd(m,n)=1.

Regards,
Max