Expansions In Non-integer Bases

The thesis mainly deals with an over view of number theory problem which grewout of the uniqueness of expansions.(1) We research the characteristics and the numberof expansions in non-integer bases.(2) x belonging to J q=[0, m (q1)]has differentexpansions in different bases. We describe the unique sets by the lexicographiccharacterizations. We will also extend the one-dimensional expansions tomulti-dimensional expansions. In ergodic theory there can be iterative map, there aretwo-dimensional or multi-dimensional unique sets and we only set focus on theirtopological properties, Hausdorff dimension and Lebesgue measure. The paper shows anumber of fundamental but powerful proofs and many examples. This article overviewsand expands related contents of the expansions in non-integer bases based on full accessto relevant documents. This paper consists of three independent chapters.In Chapter1, we give some research results which are based on these elementaryconcepts, some research status and this article mainly arrangement.In Chapter2, we give some elementary concepts which will be used in this article,such as β expansions, Hausdorff dimension, lexicographic sequences, universalexpansion, etc.In Chapter3, firstly, for a certain real number x, we give classifications which arebased on the different number of expansions in different bases. We will especially discussthe number of expansions of x=1in a given base q. Secondly, we give the numberwhich has the universal expansions in given bases and the condition that the points ofJ qwhich has universal expansions. Then we introduce the set composed by the baseswhich made that x=1has unique expansion and the set’s topological properties, wealso come to some conclusions by means of analogy with the Cantor set. Then in givenbases, we will give the properties of the set which consists of x that has unique expansion. Thirdly, we will discuss the Hausdorff measure and topological properties ofunivoque set. Finally, we generalize the alphabet to general alphabets. We will discussthose properties of expansions which are based on the general alphabets.