Run-length Function Of The Bolyai-Rényi Expansion Of Real Numbers

The Bolyai-Renyi expansion of a real number x ∈[0,1)is obtained by iterating the following mapping in the unit interval:T(x)=(x+1)2(mod 1).Through the Bolyai-Renyi transformation,almost every x ∈[0,1)can expressed in the following form (?)with digits xn ∈ {0,1,2} for all n ∈ N.For any real number x ∈[0,1)and digit i ∈ ｛0,1,2},let rn(x,i)be the maximal length of consecutive i’s in the first n digits of the Bolyai-Renyi expansion of x.In this paper,we study the asymptotic behavior of the run-length function rn(x,i).We prove that for any digit i ∈ {0,1,2},the Lebesgue measure of the set (?) is 1,whereθi=1+(?).We also obtain that the level set (?) is full Hausdorff dimension for any 0≤α≤+∞.