  Under certain natural sufficient conditions on the sequence of uniformly
bounded closed sets $E_k\subset\mathbb{R}$ of admissible coefficients, we
construct a polynomial $P_n(x)=1+\sum_{k=1}^n\varepsilon_k x^k$,
$\varepsilon_k\in E_k$, with at least $c\sqrt{n}$ distinct roots in $[0,1]$,
which matches the classical upper bound up to the value of the constant $c>0$.
Our sufficient conditions cover the Littlewood ($E_k=\{-1,1\}$) and Newman
($E_k=\{0,(-1)^k\}$) polynomials and are also necessary for the existence of
such polynomials with arbitrarily many roots in the case when the sequence
$E_k$ is periodic.
