Lebesgue Measure And Hausdorff Dimension Of Univoque Sets For ?>1

Arbitrarily given x>0,take ?>1,a sequence(xi)= x1x2… is called an expansion of x in base ?,if(?)holds,where xi?{0,1,…,[?]},[?]represents the largest integer that is less than ?.The univoque set of x,denoted by U(x),is defined as the set of all the number ?>1 such that x has a unique expansion in base ?.In this paper,we consider the Lebesgue measure and the Hausdorff dimension of U(x)as well as its topological properties.In Chapter 1,we introduce some background of ?-expansion and the significance of this research.And we list the main three theorems at the end of this chapter.In Chap-ter 2,we introduce the Hausdorff dimension,some basic properties of quasi-greedy expansions,the criterion of unique expansion,projection mapping and coding map-ping,and the Lebesgue density theorem,which are necessary to the proof of our main theorems.What's more,we proved the Hausdorff dimension of U(x)is equal to 1 in section 2.1 according to the monotonicity of Hausdorff dimension.In Chapter 3,we proved U(x)is a Lebesgue null set by using Lebesgue density theorem.In Chapter 4,we showed,by a constructivity method,that there exist strictly increasing sequences in U(x)which converge to any integer larger than 1 if x?(0,1),and there exists a strictly increasing sequence in U(x)which converges to 1/x+1 if x?(1,(?)+1/2).