Analytical Properties Of A Class Of Cantor Integers And Interval Densities Of Corresponding Cantor Sets

In this thesis,we study a class of Cantor integers ｛C_n}n≥1 with the base conversion function f:{0,…,m}→{0,…,p} being strictly increasing and satisfying f（0）=0 and f（m）=p.This thesis is divided into two parts.In the first part,let α=log_m+1^p+1.For the general base conversion function,we first present an algorithm for computing the supremum and infimum bounds of the sequence{C_n/n^α}n≥1.When the base conversion function is a quadratic function satisfying some conditions,we get the concrete value of its supremum and infimum.Then we show that{C_n/n^α}n≥1 is dense in a closed interval with the endpoints being its inferior and superior respectively.Therefore,we obtain the point which attains the maximal interval density of the interval[0,x]with respect to the self-similar measure supported on C,where C is the Cantor set induced by Cantor integers {C_n}n≥1.This result partially confirms a conjecture of Ayer and Strichartz in 1999 about the maximum interval density of[0,x].In the second part,we consider the limit functions induced by Cantor integers{C_n}n≥1.First,we prove that the limit function satisfies the（α-δ）-Holder condition on the set of（m+1）normal numbers,where δ is an arbitrarily small number in（0,1）,which shows that the limit function is differentiable almost everywhere.Finally,we give the decay rate of the logarithmic Fourier coefficient of the limit function.