The Approximation Of Complex By Eisenstein Rationals And The Inhomogeneous Diophantine Approximation Over The Field Of Formal Series

The development of the theory of Diophantine approximation started two centuries ago. It is an important branch of the number theory and its center question is that for a real number x given, to which extent can x be approximated by the rational numbers whose denominators are less than some natural number q₀, that is, how small can the difference |x —p/q| get, where p/w ∈ Q and q ≤ q₀. The classical Dirichlet theorem(in 1842) pointed out: for any real number x, there exists infinite numbers of positive integers q and integers p such thator equivalentlyholds for infinite of positive integers q, where ||y|| = min |y-m| denotes the distance from the nearest integer to y. This result may be thought of as the first one in the metric theory of Diophantine approximation.Early in the last century, Khintchine proved that if Ψ(q) is non-increasing and 0 ≤ Ψ(q) ≤ 1/2, then the set, in which any point x satisfies that there are infinite numbers of positive integers q such thatholds, is either full or null (in one-dimensional Lebesgue measure sense) accordingto the series Ψ(q) diverges or converges, that is 0 - 1 law holds for the set above.This conclusion is just the classical Khintchine theorem , which builds the foundation of the metric theory of Diophantine approximation. In the theory of Diophantine approximation , there exist many sets satisfying the 0 — 1 law. Measure-zero set is also called exceptional set, and usually it is fractal. Then how to distinguish theseexceptional sets? This is a natural question, the Hausdorff dimension provides a strongly efficient tool for us. We say that x is v—well approximable (v > 0), if there exist infinitely many positive integers q such that\\qx\\ < kl-Clearly by the Khintchine theorem, if v > 1, then the set of all v—well approximable points is exceptional. Moreover, by Jarnik theorem, we know that its Hausdorff dimension is —^.With the development of the theory of Diophantine approximation in the real numbers field, some research in other fields appears, such as in the fields of formal series, the p-adic field and so on. In this paper, we'll devote ourselves to the Eisenstein rational approximable in the complex plane and the inhomogeneous Diophantine approximation in the field of formal series. The framework of this paper is as follows:In the second chapter, some backgrounds and known results of Diophantine approximation and fractal geometry are introduced.In the third chapter, we discuss the approximation of complex numbers by the Eisenstein rationals and obtain the Hausdorff dimension of the sets of badly approximable and v—well approximable (v > l).The Eisenstein integers, are numbers of the form a + bu, where a and b are normal integers,a; = \(—1 + iy/3), is one of the roots of z3 = 1. The setZ[a;] = {a + buj : a, b e Z} is a ring, and called the ring of Eisenstein integers. The set of Eisenstein rationals is denoted by Q(u}) = {| + fa : |, 2 G Q}â– . We call the approximation of the elements of C by the elements of Q(o;) the Eisenstein rational approximation of complex numbers. We call the setB(u) = {z € C : 3K = K(z), \z-^\> pp V^ € Q[ is the set of badly approximable and the setWw(w) = izeC: z-- < \q\^v+1 holds for infinitely many- e Q(w) 1 I 9 9 Jis the set of v—well approimable.In the forth and final chapter, we development the theory of inhomogeneous Diophantine approximation in the field of formal series, we obtain the analogues of Khintchine theorem and Jarnik-Besicovitch theorem. Let F be a finite field, X is an indeterminate, F((X¹)) denotes the field of formal of formal series with coefficients from F, j.| is a non-Archimedean value over this field. Suppose I — {x G F((X¹)) : |x| < l}.We denote ï¿¡1 = I x I. Forq e F[X], let* V[X] -* R+be a function satisfies | andwithout confusion , let*(g) = *(|<7|). We call the set$(*]/) = {(^ a) Etl: \\qx - a\\ < V(q) holds for infinitely many? e F[X]}is the set of *—approximate in Q, the approximation over above is called inhomogeneous Diophantine approximation. We call the set$? = {(x,a) E Q : \\qx - a\\ < \q\^v holds for infinitely many? G F[X]}is the set of v—approximable in Q,. At first, we prove the set of *—approximable obey 0 — 1 law (in sense of 2-dimensional Haar measure) according to the series diverges or converges, where we need not limit the monotone of \I>. Thus,Q€$[X]the set <&? is full when v < 1. if v > 1, the set $? is exceptional set. Moreover, if is non-increasing, we adapt a technique using so-called Ubiquitous systems(U-system), and we obtain the Hausdorff dimension of the set of $(*). For q 6 F[X], let ty(q) = |?|^{^}+1\ in this way we also obtain the Hausdorff dimension of the set...