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For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). Let $R_n(t)$ denote the number of real roots of $\sum_{1\leq k\leq n}\epsilon_k(t)z^k$. Is it true that, for almost all $t\in (0,1)$, we have \[\lim_{n\to \infty}\frac{R_n(t)}{\log n}=\frac{\pi}{2}?\]
Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.

See also [522].