  We investigate properties of the $m$-th error of approximation by polynomials
with constant coefficients $\mathcal{D}_{m}(x)$ and with modulus-constant
coefficients $\mathcal{D}_{m}^{\ast}(x)$ introduced by Bern\'a and Blasco
(2016) to study greedy bases in Banach spaces. We characterize when
$\liminf_{m}{\mathcal{D}_{m}(x)}$ and $\liminf_{m}{\mathcal{D}_{m}^*(x)}$ are
equivalent to $\| x\|$ in terms of the democracy and superdemocracy functions,
and provide sufficient conditions ensuring that $\lim_{m}{\mathcal{D}_{m}^*(x)}
= \lim_{m}{\mathcal{D}_{m}(x)} = \| x\|$, extending previous very particular
results.
