The empty space in the middle is pretty easy. These are roots of polynomials of the form +- z^n +- z^(n-1) +- ... +- 1. If |z| is small then the earlier terms a

The empty space in the middle is pretty easy. These are roots of polynomials of the form +- z^n +- z^(n-1) +- ... +- 1. If |z| is small then the earlier terms are smaller; the absolute value of their sum is at most |z|^n + ... + |z|^1 which is at most |z| / (1-|z|). If this is < 1 then the sum can't be large enough for adding it to +- 1 to give 0. So if |z| < 1-|z|, i.e., if |z| < 1/2, then z can't be a root. That's for the "main" image. The first image in Baez's post is for polynomials whose coefficients are -4,-3,...,+4, and the analysis would be different for those, but it's still true that if |z| is small enough then the sort of calculation in the previous paragraph forces it not to be the root of any nonzero polynomial with small integer coefficients. The holes near +- 1 are more complicated, I think.