A note on approximation efficiency and partial quotients of Engel continued fractions

We consider the efficiency of approximating real numbers by their convergents of Engel continued fractions (ECF). Specifically, we estimate the Hausdorff dimension of the set of points whose ECF-convergents are the best approximations infinitely often. We also obtain the Hausdorff dimensions of the Jarnik-like set and the related sets defined by some growth rates of partial quotients in ECF expansions.