Sorry. I mean that there is a uniform constant C > 0 such that all representations of 0 with a_0=1 and a_i \in {-1,0,+1} have density at least C (in all initial subsequences). Cris On Oct 15, 2013, at 2:56 PM, rcs@xmission.com wrote:
Cris, you're silent on uniformity: Do you want "there's a positive C such that all representations of 0 with leading digit +1 have lim inf (%nonzero digits) >=C", or "for each representation R ... there's a positive C(R) such that ..." but maybe the R's can approach 0?
Rich
----- Quoting Cris Moore <moore@santafe.edu>:
If we're talking about fractional bases, allow me to introduce you to a conjecture of Leonard J. Schulman and me:
Consider writing zero in base 3/2. That is, for i=0,1,2... let a_i be an integer, and let
0 = \sum_{i=0}^\infty a_i (2/3)^i .
Furthermore, assume that a_i is in {+1,0,-1} for each i, and that a_0 = 1. Thus the first term is 1, and the remaining terms need to cancel it out.
Conjecture: all initial subsequences have a nonzero density of nonzero coefficients. That is, there is a constant C > 0 such that, for every n, at least Cn of the coefficients a_1, a_2, ... a_n are nonzero.
I am equally interested in any nearby conjecture: e.g. replace 3 with 5 or 7, let a_i belong to {-2,-1,0,1,2}, and so on.
If you can prove this, it will have nice consequences for a construction in computer science called tree codes: see http://arxiv.org/abs/1308.6007
- Cris