Oh, the "greedy representation" I was thinking about wasn't focused on zeros specifically -- rather it just appended whichever of {-1,0,+1} brought the partial sum closer to the target. (Presumably in case of a tie you would append a 0.) There's also the "greedily use 0's" representation, in which your next digit is a 0 unless that would render the goal unreachable. --Michael On Sat, Oct 19, 2013 at 11:55 AM, Cris Moore <moore@santafe.edu> wrote:
I suppose the greedy representation is to set a_i = +1 or -1 depending on whether the sum we have so far is positive or negative. But we can get away with a lot of zeros if we achieve a partial sum that's quite close to zero... we don't need to use nonzero a_i until we get nervous about whether the terms are decreasing too fast in absolute value to get to zero from the current partial sum.
Cris
On Oct 19, 2013, at 8:54 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
Sorry I'm late to this game, just trying to wrap my head around balanced base 3/2. Can you explicitly write down any nontrivial representation of 0? Should it be obvious to me what the "greedy" representation of 0 starting with a_0=1 is?
Clearly I should read your paper, though this tastes enough like elliptic theta that I'm already a little scared.
--Michael
On Tue, Oct 15, 2013 at 5:31 PM, Cris Moore <moore@santafe.edu> wrote:
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
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