In this dissertation,we concern with the distribution of a class of integer sequences which are closely related to Cantor sets.For any positive integer p≥3,the Cantor sets can be constructed by limiting the range of the digits in the p-ary expansion of real number in[0,1]:let A={τ0,…-,τs-1} be a proper subset of {0,1,…,p-1},the Cantor set above is defined as C={∑k≥1εkp-k:εk∈A(k≥1)}.Especially,C is the classic middle third Cantor set if p=3,A={0,2}.Similarly,an integer sequence can be obtained by limiting the range of the digits in the p-ary expansion of integers:{an}n≥0={∑i=0kεipi:εi∈A(0≤i≤k),k∈N},which is called Cantor integer sequence,and the elements in it are called the Cantor integers.Including the first chapter of introduction and the second chapter of preliminary,there are five chapters in the thesis.In the third chapter,aided by the relation of the Cantor integer sequence {an}n≥0 and the Cantor set c,we show that the growth order of the arithmetic function an is logsp,where s is the number of elements in A.In view of the fact that C can be designated as a self-similar set,which is the attractor of the family {Si}0≤i≤s-1 on[0,1],where Sy(x)=x/p+τi/p.We establish a clear connection between {an/nlogsp}n≥1 and the self-similar measure μc on c with μc=1/s∑i=0kμcSi-1,which says that the set{x/(μc[0,x]))logsp:x∈c∩[f(1)/p,1]} is precisely the set of the accumulation points of{an/nlogsp}n≥1.We show that inf{{an/nlogsp:n≥1}=q(s-1)+r/p-1,sup{an/nlogsp:n≥1}=q(p-1)+pr/p-1,and {an/nlogsp}n≥1 is dense in[q(s-1)+r/p-1,q(p-1)+pr/p-1],if there exist q≥2,r ∈{0,…,q-1},such that for any i ∈{0,…,s-1},ai≡r(mod q).The distribution of {an/nlogsp}n≥1 was probed into further in the fourth Chapter.We show that if {an/nlogsp}n≥1 is dense in[m,M],then {an/nlogsp}n≥1 does not have the cumulative distribution function,but has the logarithmic distribution function(given by a specific Lebesgue integral).In the last chapter,we conclude the main research result of this thesis and broach the further research. |