Define density $ (f (z)) $ of a power series $ f (z) = f_0 + f_1z + f_2z ^ 2 + … $ in the binary power series ring $ F_2 ((z)) $ as the natural density of the whole $ E_f: = $ {$ i: f_i = 1 $}

$ D: = $ {$ (f -1): 0 $ in $ E_f $} is the set of densities of the reciprocals of the polynomials with a constant nonzero term.

I think every arithmetic sequence has the form $ E_ {f <-1} $ where $ f (z) $ is a polynomial with $ f (0) = 1 $

1) Any idea about whether $ D $ Should it be dense anywhere in (0,1)?

2) Any idea of ββthe upper limits for $ D $ ? inf$ (D) = 0 $

I remember that part of this arose in the "Binary Power Reciprocals" series by Cooper and Bryant led by questions about the parity of the partition function and, among other things, the densities of the polynomial reciprocals.