Fractional dimension of some exceptional sets in continued fractions

In this paper, we calculate the Hausdorff dimension of some exceptional sets that emerge from specific constraints imposed on the partial quotients of continued fractions. In particular, we calculate the Hausdorff dimension of the sets ({l Xn 11111 A1 = x E (0,1) : an+1(x) >= ai ( x), for all n E N , i=1 and ({l Xn 11111 A2 = x E (0,1) : an+1(x) >= ai ( x), for infinitely many n E N . i=1 We prove that the Hausdorff dimensions of A1 and A2 are 1/2 and 1 respectively. The Hausdorff dimension of some other related sets, obtained by considering different faster growth rates such as replacing the growth rate of sums of partial quotients with the product of partial quotients in the above sets, is also calculated with the dimension bounds 1/3 and at least 2/3.