HAUSDORFF DIMENSION FOR THE SET WITH EXPONENTIALLY GROWING PARTIAL QUOTIENTS IN THEIR LUROTH EXPANSION

For any x E (0, 1], let [d(1)(x), d(2)(x),...] be its Lu center dot roth expansion. We prove the exact Hausdorff dimension of the set defined as {x is an element of (0, 1] : ciAn i < dn+i(x) < 2ciAni , 0 < i < m - 1, for infinitely many n is an element of N} , where each partial quotient grows exponentially and the base is given by the parameters Ai > 1, and ci > 0 are fixed real numbers. This set generalises several known sets and by fixing specific choices of parameters we obtain lower bounds for the Hausdorff dimension for various limsup sets. In particular, we obtain lower bounds for the Hausdorff dimension of the following sets: E-m(B) := {x E (0,1] : m -1 pi i=0 dn+i(x) > Bn infinitely many n E N}, and ( )dn(x)dn+1(x) > Bn1 for infinitely many n E N FB1,B2 = x E (0, 1] : , dn+1 (x) < B2n for all sufficiently large n E N where m E N, 1 < B, B-1, B-2 <