Some Studies On ?-expansions And Continued Fractions

This thesis focuses on small bases which admit countably many expansions with mul-tiple digits,small bases for which 1 has countably many expansions with multiple digits,and maximal continuant of singular continued fraction in the ternary case.We calculate the second smallest bases for the former two issues respectively and provide the unique arrangements attaining the maximal continuant of singular continued fraction.It consists of six chapters.In the first chapter,we introduce the background.In the second section,we give the definition and some elementary properties.And in the next three sections,we discuss the three problems in details.In the third chapter,given a positive integer M,let BN0(M)be the set of bases q>1 such that there exists a real number x with exactly N0 different q-expansions over the alpha-bet {0,1,...,M}.It is known that the smallest base in BNO(M)is G(M),the generalized golden ratio.In this chapter we investigate the next smallest element qN0(M)of BN0(M),and show that if M=2m,qN0(M)is the appropriate root of g3=mq2+(m+1)q+1,and if M=2m-1,qN0(M)is the appropriate root of q6=(m-1)q5+(2m-1)q4+(2m-1)q3+2mq2+mq+1.In the fourth chapter,given a positive integer M,let B1,N0(M)be the set of bases q>1 such that 1 has exactly N0 different q-expansions over the alphabet{0,1...,M}.It is known that the smallest base in B1,N0(M)is G(M).In this chapter we investigate the next smallest element q1,N0(M)of B1,N0(M)is q3(M),the appropriate positive root of q3-(m+1)q2-q+1=0 if M=2m and the appropriate positive root of q5-mq4-mq3-q+1=0 if M=2m-1.In the fifth chapter,a regular(resp.singular)continuant is the denominator of the convergents of the regular(resp.singular)continued fractions.Nicol posed the following problem:given any finite sequence of positive integers,find the permutations whose contin-uant attain the extremal values.The answers are known for the regular continued fraction expansion,as well as the minimal continuants in the singular expansions.While only partial results exist for the case of maximal singular continuants,namely,the cases that the pairwise distinct entries,and the sequences consisting of two digits.In this chapter,we elucidate the unique arrangements of the maximal singular continuants for the sequences consisting of three letters by an iterated algorithm.In the last chapter,we conclude the main results,and give some topics for future research.