Limit Theorems And Dimension Studies In Continued Fractions And β-expansions

The dissertation studies the limit theorems and dimensional properties in continued fractions and β-expansions,the main results are summarized as follows:1)We study the Hausdorff dimension of the set related to maximal partial quotients Mn(x)and the large deviation for the denominator qn(x)of the convergent.Mn(x)is an important quantity in studying partial quotients.The existed results focus on the Hausdorff dimensions of the sets of points x for which the maximum Mn(x)increases to infinity with polynomial rates and one-fold exponential rates.We are concerned with the case of two-fold exponential rates and obtain that there is an inverse relation between the Hausdorff dimension of the corresponding set and the second base in the two-fold exponential function.The result can be treated as an important complement to the earlier studies.Moreover,if one considers that Mn(x)increases to infinity with a rate faster than the two-fold exponential rate,then the corresponding set is of Hausdorff dimension zero.It means that the result is final in the view of Hausdorff dimension.qn is a quantity closely related to the Diophantine approximation.The existed studies focus on the asymptotic behavior,central limit theorem,law of the iterated logarithm,etc.,but these results do not relate to the order of convergence.Estimating the order of convergence is a very important problem in the error estimate and accuracy analysis.We obtain that the speed of convergence in measure of(log qn(x))/n is exponential by establishing the relation between the asymptotic behavior of E(qnθ)and the function P(θ)and using the methods of large deviations.2)We are concerned with the approximation orders of real numbers by β-expansions.For any number x ∈[0,1),the sum of the first n terms in the β-expansion of x is called the convergent denoted by ωn(x).First,we prove that ωn(x)converges to x with exponential order β-n for almost all x in the sense of Lebesgue measure.A natural question is if there exists some points such that ωn(x)converges to x with other orders.If yes,how large is the set of such points.We obtain:there is no point such that its convergets converge to itself with the order β-αn(0≤α<1);the set of x’s for which ωn(x)converges to x with the order β-αn(α>1)is at most countable;the set of x’s for which ωn(x)converges to x with the order β-n(α>1)of course has Lebesgue measure.Further,we successfully apply these results to investigate the asymptotic behavior of the orbits of real numbers under β-transformation,the shrinking target problem for β-transformation,the Diophantine approximation problem for β-expansions and the properties of run-length function.3)We discuss the relation between continued fractions and β-expansions.For any irrational numbers x ∈[0,1)and positive integers n,define kn(x)= sup {m ≥ 0:J(ε1(x),…,εn(x))(?)I(a1(x),…,am(x))},where J(ε1(x),…,εn(x))and I(a1(x),…,am(x))respectively stand for the cylinders of order n and of order m in the β-expansion and the continued fraction expansion of x.When β = 10,Lochs,Faivre,Wu respectively obtained the asymptotic behavior,large deviation,central limit theorem,law of the iterated logarithm of kn(x).It is worth pointing out that the proofs of these results heavily rely on the length of cylinders in decimal expansion.When β is integer,the cylinder of order n is a left-closed and right-open interval and its length constantly equals to β-n;When β is non-integer,the cylinder of order n is irregular and the lower bound of its length can be much smaller than β-n.So the methods for kn(x)in decimal case can not be applied to obtain the above results of kn(x)in the general case.First we give an appropriate estimation for the lower bounds of cylinders;then study the distribution and limit behavior of the lower bound,and prove the large deviation,central limit theorem,law of the iterated logarithm of kn(x).Besides,we also obtain that the order of convergence for the approximation of a real number byβ-expansions better than the approximation by continued fractions or the approximation of a real number by continued fractions better than the approximation by β-expansions.At last,we give the large deviation principle for Engel continued fractions and its rate function.