SOLVED
Is it true that all except $o(2^n)$ many polynomials of degree $n$ with $\pm 1$-valued coefficients have $(\frac{1}{2}+o(1))n$ many roots in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$?
Erdős and Offord
[EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.
Solved by Yakir [Ya21], who proved that almost all such polynomials have
\[\frac{n}{2}+O(n^{9/10})\]
many roots in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$.
See also [474] and [521].
