Estimates for the wandering rate of solutions of a linear differential equation via its coefficients

For nontrivial solutions of a linear nonautonomous differential equation with integrally small coefficients, we improve earlier-known upper bounds for the wandering rate. In particular, our estimates imply that the upper bound of the range of the wandering rate for equations of arbitrary order tends to zero as all of their coefficients uniformly (on the time half-line) tend to zero at infinity.