From: "TwoBirds"
Subject: Re: "periodicities" in characteristics
Date: Fri, 5 Mar 1999 19:41:42 -0800
Newsgroups: sci.math
Keywords: relationship between continued fractions and c(k)=[(k+1)x]-[kx]-[x]
John R Ramsden wrote ...
>TwoBirds wrote ...
[...]
>>Suppose x is irrational with simple continued fraction [a(0);a(1),a(2),...],
>>having the convergents p(n)/q(n) = [a(0);a(1),a(2),...,a(n)], q(0)=1,
>>and whose characteristic is the binary string c(1)c(2)c(3)...,
>>where c(k) = [(k+1)*x] - [k*x] - [x], k=1,2,3,... , and []=floor.
>>[...]
>>I conjecture that the characteristic of any irrational has the structure
>>[*1] c(1) c(2) c(3) ... = s(1)^a(1) s(2)^a(2) s(3)^a(3) ...
>>[*2] length(s(k)) = q(k-1), k=1,2,3,...
>>and I want to find the explicit form of s(k), k=1,2,3,...
[...]
>>In fact, I've been able to deduce from a theorem of Markov that [*] is
>>correct at least up to the third factor, with
>>
>>s(1) = 0
>>s(2) = 1 s(1)^(a(1)-1)
>>s(3) = 0 s(2)^a(2)
>>
>>but I don't know the explicit form of s(k) for k>=4.
>
>"Elementary Number Theory" by Venkov has a longish section on this.
>It may include what you are looking for (or more than you've found so far).
Thank you for the reply.
Venkov was my source for the "theorem of Markov" mentioned above.
(That longish section you noted is Venkov's proof of that theorem, which
is indeed a recursion for the characteristic, but it's not in the form I
have
conjectured. Venkov/Markov's recursion is so different that I was surprised
to be able to manipulate it into the form above for s(1), s(2), s(3).
But s(4) etc looked way too daunting, and I can see no way to show the
equivalence in a general way that would automatically provide the form of
s(k), k=1,2,3,... .