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 ... ... ... ... -----