Multifractal Analysis of Convergence Exponents for Products of Consecutive Partial Quotients in Continued Fractions

For each real number x is an element of(0,1), let [a1(x),a2(x),<middle dot><middle dot><middle dot>,an(x),<middle dot><middle dot><middle dot>] denote its continued fraction expansion. We study the convergence exponent defined by tau(x) := in f{s >= 0 :(infinity)& sum;(n=1)(an(x)an+1(x))-s<infinity}, which reflects the growth rate of the product of two consecutive partial quotients. As a main result, the Hausdorff dimensions of the level sets of tau(x) are determined.