On an error estimate for the averaging method in a two-frequency problem

The author obtains an unimprovable estimate of the averaging method for a two-frequency problem with analytic right-hand sides under condition $\overline A$, which means a nonzero rate of change of the frequency ratio along trajectories of the averaged system. It turns out to be of order $\varepsilon^{\frac14+\frac1{2(l+1)}}$ for initial data outside a set of measure of order $\varepsilon^{\frac12}$, where $\varepsilon$ is a small parameter of the problem and $l$ is an upper bound for the maximal multiplicity of the roots of a certain finite set of equations (it is assumed that $l>1$).
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