@aa82d4be - cool, I'm glad you're spreading the news about this wonderful set.

Our paper says what people have proved, or at least what we know they've proved:

https://www.ams.org/journals/notices/202309/rnoti-p1495.pdf

It's known that the set of all roots of all polynomials with coefficients ±1 is contained in the annulus 

1 ≤ |z| ≤ 2 

and that it contains the annulus

2^(-1/4) ≤ |z| ≤ 2^(1/4)

It's also known that this set is connected and locally path-connected!  But that's about all.

So given what people have proved so far, in theory the set could fill in the whole annulus 1 ≤ |z| ≤ 2.  But from numerical computations I'm completely sure that's not true.  So, someone should start proving theorems about this stuff!