The paper is concerned with the construction of certain Euclidean-like coordinate systems at infinity of complete Riemannian manifolds with curvature decay (for the Riemann and Ricci curvature tensors) and volume ascent of balls of prescribed order \(<-2\) and n, respectively. In particular, a conjecture of \textit{H. Nakajima} on Ricci-flat manifolds of dimension \(\geq 4\) can be answered with these methods [J. Fac. Sci., Univ. Tokyo, Sect. I A 35, 411-424 (1988; Zbl 0655.53037)]. The proofs are highly technical, consisting of numerous estimates, and relying on results on Laplacian and Hessian comparison theorems, harmonic coordinates, Gromov's convergence theorem and J. Moser's iteration scheme.