Then they won't span the whole vector space. I think you need something closer to Dirichlet characters in order to get a nice basis: https://en.wikipedia.org/wiki/Dirichlet_character -- APG.
Sent: Saturday, June 23, 2018 at 3:05 AM From: "Tomas Rokicki" <rokicki@gmail.com> To: "Dan Asimov" <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Constructing a set that isn't a congruence class
In order to make those bit strings linearly independent, you should only include prime periods. But this messes up the predictability of your diagonal . . .
On Fri, Jun 22, 2018 at 5:34 PM Dan Asimov <dasimov@earthlink.net> wrote:
These things form a very interesting set.
Conjecture (either vague enough to not be controversial, or dead wrong): ---------- If we think of these bit strings as vectors in the vector space V that is a countably infinite direct product of the 2-element field:
∞ V = ∏ Z/2 1
then every sequence
b : Z+ —> {0,1}
of bits is the sum of elements {E_(n_j) | j in Z+} on Jim's list, i.e., where each element occurs at most once. (Since only finitely many elements have any 1's among the first n bits, any such countable sum will make sense.)
Now I'm wondering if Jim's list of pure frequencies among the periodic bit strings can also be seen as generating — *in some sense* — all sequences of integers, or of positive integers.
Questions: ----------
I) What are the sets of sequences of arbitrary integers
x : Z+ —> Z
of the form
{x = Sum N_j E(j), N_j in Z}
and
II) What are the sets of sequences of positive integers
of the form
{x = Sum N_j E(j), N_j in Z+}
—Dan
Jim Propp wrote: ----- There are only countably many sets of natural numbers that form (the nonnegative part of) a congruence classes. Here's a natural way to list them as bit strings:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ... 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ... 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ... 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 ... 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 ... 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 ... 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 ... 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 ... 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 ... 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 ... 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ... 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ... 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 ... 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 ... 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 ... 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 ... ... ... ... -----
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