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When analyzed in terms of the Symanzik expansion, lattice correlators ofmulti-local (gauge-invariant) operators with non-trivial continuum limitexhibit in maximally twisted lattice QCD ``infrared divergent'' cutoff effectsof the type a^{2k}/(m_\pi^2)^{h}, 2k\geq h\geq 1 (k,h integers), which tend tobecome numerically large when the pion mass gets small. We prove that, if theaction is O(a) improved a` la Symanzik or, alternatively, the critical masscounter-term is chosen in some ``optimal'' way, these lattice artifacts arereduced to terms that are at worst of the order a^{2}(a^2/m_\pi^2)^{k-1}, k\geq1. This implies that the continuum extrapolation of lattice results is smoothat least down to values of the quark mass, m_q, satisfying the order ofmagnitude inequality m_q >a^2\Lambda^3_{\rm QCD}.Comment: 23 pages, no figures. Revised version, published by JHEP. Various  sections shortened, alternative determination of the "optimal" critical mass  discussed, section about the pion decay constant inserted. Conclusions  unchange

Reducing cutoff effects in maximally twisted LQCD close to the chiral limit