The Newman algorithm for constructing polynomials with restricted coefficients and many real roots

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 $P_n(x)=1+{\sum }_{k=1}^n{\epsilon }_k{x}^k$, ${\epsilon }_k\in E_k$, 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.