ON ARITHMETIC PROGRESSIONS IN THE DIGITS OF ENGEL EXPANSIONS

Denote by [Formula: see text] and [Formula: see text] the sets of real numbers whose Engel expansion digits contain arbitrarily long arithmetic progressions and arbitrarily long arithmetic progressions with arbitrary common difference, respectively. In this paper, the sizes of [Formula: see text] and [Formula: see text] are investigated from the measure-theoretical, fractal and topological viewpoints. We prove that [Formula: see text] and [Formula: see text] are of Lebesgue measure zero but have full Hausdorff dimension. Moreover, [Formula: see text] is of first category while [Formula: see text] is residual.