On Small Bases For Which 1 Has Countably Many Expansions

This paper investigates the small bases for which 1 has countably many q-expansions for q∈(1,2).This paper includes five chapters.The first chapter prepares for introduction,some results of this paper are also included. The second chapter prepares for some preliminary knowledge.The main contents of the next two chapters are in the following.Let q ∈(1,2),we consider the q-expansion(δi)i=1∞ of number x in Iq:= [0,1/(q-1)] of the form with δi ∈{0,1}.Let and Here and #∑q(x)denotes the cardinality of the set ∑q(x).Sidorov and Vershik showed in[19]that min BN0=min B1,N0=(1+1/2/5)/2,and Baker showed in[1]that BN0 ∩ ({1+(1/2)/5}/2,q1]={q1},where q1(≈1.64541)is the positive root of x6-x4-x3-2x2-x-1=0. In this paper we show that the second smallest point of B1,N0,is q3(≈1.68042),the positive root of x5-x4-x3-x+1=0. Enroute to proving this result we show that BN0 ∩(q1,q3]={q2,q3},where q2(≈1.65462)is the positive root of x6-2x4-x3-1=0.The final chapter is devoted to summarizing the results of this dissertation and giving some questions.