ON SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS AND HAUSDORFF DIMENSIONS

For any x is an element of (0, 1], let the infinite series Sigma n=1 infinity</mml:msubsup> <mml:mfrac>1 d1(x)d2(x)dn(x)</mml:mfrac> be the Engel expansion of x. Suppose psi : N -> + is a strictly increasing function with limn -> infinity psi (n) = infinity and let E(psi), Esup(psi) and Einf(psi) be defined as the sets of numbers x is an element of (0, 1] for which the limit, upper limit and lower limit of <mml:mfrac>log dn(x) psi (n)</mml:mfrac> is equal to 1. In this paper, we qualify the size of the set E(psi), Esup(psi) and <mml:msub>Einf(psi) in the sense of Hausdorff dimension and show that these three dimensions can be different.