On A Class Of Fractal Sets In Lüroth Expansion

Let(?)be the dyadic expansion of x∈[0,1),where εn(x)∈{0,1} is called the n-th digit of the dyadic expansion of x.The asymptotic behavior of the run-length functionrn(x)= max{k:εj+1(x)=…= εj+k(x)= 1,for some 0 ≤ j ≤n-k}in dyadic expansion was first studied by Erdos and Renyi,they showed that(?)where L denotes the 1-dimensional Lebesgue measure.For an infinite symbolic system-Liiroth dynamical system,Sun and Xu showed that the maximal run-length function(?)in the Lüroth expansion enjoyed the following large number law,(?)where(dn(x))n≥1 is the digit sequence of the Lüroth expansion of x:(?)This dissertation is devoted to study a class of exceptional sets(?)induced by the above large number law.We will give an estimation of the upper bound of the Hausdorff dimension of F(β).