Reducing cutoff effects in maximally twisted lattice QCD close to the chiral limit

When analyzed in terms of the Symanzik expansion, the expectation values of multi-local (gauge-invariant) operators with non-trivial continuum limit exhibit in maximally twisted lattice QCD ``infrared divergent'' cutoff effects of the type a^{2k}/(m_\pi^2)^{h}, 2k\geq h\geq 1 (k,h integers), which tend to become numerically large when the pion mass gets small. We prove that, if the critical mass counter-term is chosen in some ``optimal'' way or, alternatively, the action is O(a) improved {\it \`a la} Symanzik, these lattice artifacts are reduced to terms that are at worst of the order a^{2}(a^2/m_\pi^2)^{k-1}, k\geq 1. This implies that the continuum extrapolation of lattice results is smooth at least down to values of the quark mass, $m_q$, satisfying the order of magnitude inequality m_q >a^2\Lambda^3_{\rm QCD}.