Some Sets Arising In Metric Number Theory

This article focuses on the efficiency of approximating real numbers by Luroth expan-sion, the growth speed of the digits in infinite iterated function systems, and the property of the β-expansion of the unit. We get the Hausdorff dimensions of some sets arising in such fields. The first chapter describes the background of this paper, and the preliminaries are given in chapter II. And then with three chapters, the three issues are discussed in detail.In the third chapter, we consider the efficiency of approximating real numbers by Luroth expansion. We consider the set of points whose convergents in Luiroth expansion are the optimal approximation for infinitely many times, and find that it is of Lebesgue measure zero. However, the Hausdorff dimension of this set is strictly greater than0, we estimate its lower bounded. At the same time, we also prove a result similar to Jarnik theo-rem. The Hausdorff dimension of the corresponding set will reduce by half if we replace the convergents of the continued fraction expansion by the convergents of Liiroth expansion.In the fourth chapter, the concept of the partial quotients in continued fraction is ex-tended to infinite iterated function systems, called digit. We consider the growth speed of the digits in a class of infinite iterated function systems. The results of Wang and Wu, Luczak about the continued fraction were extended to a class of infinite iterated function systems. In precise, for any infinite subset B(?)N,we determine the Hausdorff dimension of the set of points whose digits belong to B and tend to infinity. For any α,b>1, consider the set of points whose digits an(x) greater than abn for any n and the set of points whose digits an(x) greater than abn for infinitely many n. The upper bounded of the Hausdorff dimension was estimated, and an example was given to show that there does not exist a uniform lower bounded if without any additional condition. In the results above, the Hausdorff dimension of the sets is tightly connected with the convergence exponent of the contraction ratios of the infinite iterated function systems. The two results above are sharp in the sense that, for any infinite iterated function system which satisfies some conditions and any functionφ(n) which tends to infinity, an infinite subset B(?)N with the same convergence exponent can be found, such that the Hausdorff dimension of the set of points whose digits belong to B and satisfy an(x)≥φ(n) for any n is zero. Meanwhile, we also prove that for any real number s0∈[0,1] and any functionφ(n) which tends to infinity, there is an infinite iterated function system with convergence exponent equals to S0, such that the Hausdorff dimension of the set of points satisfy αn (x)>φ(n) for all n is zero. In the fifth chapter, we study the property of the orbit of1under the β transformation. We show that for any given point x0∈[0,1] and any interval (β0,β1)(?)(1,∞), the set of β∈(β0,β1) such that x0is not an accumulation point of the orbit of1under the0transformation has the full Hausdorff dimension.Finally, in the sixth chapter, we summarize the main results of this paper, and list some topics for future research.