MULTIFRACTAL ANALYSIS OF THE DIVERGENCE POINTS ASSOCIATED WITH THE GROWTH OF DIGITS IN ENGEL EXPANSIONS

In this paper, we are concerned with the multifractal analysis of the divergence points in Engel expansions. Let x is an element of (0,1) be an irrational number with Engel expansion < d(1)(x),d(2)(x),d(3)(x),...>. For any 0 <= alpha <= beta <= infinity, let D(alpha,beta):={x is an element of (0,1)\Q: liminf(n ->infinity) logd(n)(x)/log n=alpha, limsup(n ->infinity) logd(n)(x)/logn = beta}. We prove that the Hausdorff dimension of D(alpha, beta) is (alpha-1)/alpha when 1 <= alpha <= infinity , and it is zero when 0 <= alpha < 1. This indicates that the Hausdorff dimension of D(alpha,beta) is independent of beta. A very different phenomenon is shown for the gap of consecutive digits. For any irrational number x is an element of (0,1) and n is an element of N , let Delta(n)(x):=d(n)(x)-d(n-1) with d(0)(x)equivalent to 0. We derive that, for any 0 <= alpha <= beta <= infinity , the set Delta(alpha,beta):={x is an element of (0,1)\Q:liminf(n ->infinity)log Delta(n)(x)/log n=alpha, limsup(n ->infinity)log Delta(n)(x)/log n=beta} has Hausdorff dimension beta/(beta+1).