Some Fractal Questions In L(?)roth Expansions And N-continued Fraction Expansions Of Numbers

This dissertation mainly discusses some fractal questions arising in Lüroth expansions and N-continued fraction expansions of numbers.Firstly,we concern the growth property of the product of consecutive digits in the Lüroth expansion of numbers.After that,we consider the metrical theory of the partial quotients in the N-continued fraction expansion of numbers.In addition,for fractional parts of the powers of real numbers,we obtain the metrical results on the distribution of them.The whole dissertation is divided into six chapters,among which the related backgrounds and some preliminaries will be given in the first two chapters.We investigate the above several aspects specifically in the following three chapters.In the third chapter,we concern the growth property of the product of consecutive digits related to the denominator of the convergent for the Lüroth expansion of a number.For x ∈(0,1],let[d1(x),d2(x),…]L be its Liiroth expansion and(Pn(x)/Qn(x))n≥1be the sequence of convergents of the Lüroth expansion of x,where dn(x)are called the digits of the Lüroth expansion of x.Given a natural number m,the Hausdorff dimension of sets and are given.In the fourth chapter,we consider the metrical theory of the partial quotients in the N-continued fraction expansions of numbers.For any fixed positive integer N E N,every x ∈[0,1)can be expanded into an N-continued fraction expansion,denoted by x=[a1(x),a2(x,…]N,where an(x)are called the partial quotients of the N-continued fraction expansion of x.Let φ:N→R+,the Borel-Bernstein theorem and Hausdorff dimension of the set{x∈[0,1):an{x)≥φ(n)for infinitely many n ∈N｝are determined.In the fifth chapter,from a metrical point of view,we discuss the Diophantine properties of fractional parts of the powers of real numbers.Let y=(yn)n≥1be an arbitrary sequence of real numbers in[0,1]and ψ:N→R+ be a non-increasing function with ψ(n)→ 0 as n→∞.We show that there is a 0-1 law on the Hausdorff dimension of the following set E(ψ,y)={x>1:‖xn-un‖<ψ(n)for infinitely many n ∈ N｝according as lim infn→∞-logψ(n)/n=+∞ or not.Finally,the last chapter is devoted to summarizing our main results in this dissertation and proposing some questions for further study.