Hausdorff Dimension Concerning The Growth Rate Of The Digits In Alternating Oppenheim Expansions

Let x∈(0,1],under the algorithm x=x1,Dn=[1/xn]+1,xn=1/Dn-an/bn·xn+1,inducing the alternating Oppenheim series expansion of x:x=1/D1-a1/b1 1/D2+…+(-1)n a1a2…an/b1b2…bn 1/Dn+1+…,where an = an(D1,…Dn),bn=bn(D1,…,Dn)are positive integer valued functions and[y]denotes the integer part of y.In this dissertation,we study the Hausdorff dimension of the set Cm={x∈(0,1]:m≤Dn(x)/hn-1(Dn-1(x))<2m,n≥1},where m is integer and m>1.A analogous question arising from reprentations of real numbers by infinite series,was first settled by J.Galambos[1]in 1976.The main tool of our study is the proof of theorem 6,which gives the interval length that we want from the proof.Moreover,constructing skillfully set and mass distribution of J.Wu and B.W.Wang[2]is another tool.The study of this problem can provide more convenience for the research of alternating Oppenheim series expansion in the future.