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We investigate properties of the $m$-th error of approximation by polynomialswith constant coefficients $\mathcal{D}_{m}(x)$ and with modulus-constantcoefficients $\mathcal{D}_{m}^{\ast}(x)$ introduced by Bern\'a and Blasco(2016) to study greedy bases in Banach spaces. We characterize when$\liminf_{m}{\mathcal{D}_{m}(x)}$ and $\liminf_{m}{\mathcal{D}_{m}^*(x)}$ areequivalent 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 particularresults.

A remark on approximation with polynomials and greedy bases