Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II)

We prove that there is an absolute constant $c>0$ such that for every $$a_0,a_1,\dots ,a_n\in [1,M],1\le M\le \frac{1}{4}\exp \left(\frac{n}{9}\right),$$ there are $$b_0,b_1,\dots ,b_n\in \{-1,0,1\}$$ such that the polynomial $P$ of the form ${\displaystyle P(z)=\sum _{j=0}^n{b}_j{a}_j{z}^j}$ has at least ${\displaystyle c{\left(\frac{n}{\log (4M)}\right)}^{1\mathrm{/}2}-1}$ distinct sign changes in $I_a:=(1-2a,1-a)$, where ${\displaystyle a:={\left(\frac{\log (4M)}{n}\right)}^{1\mathrm{/}2}\le 1\mathrm{/}3}$. This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.