A Class Of Generalized Besicovitch Sets And Their Dimensions

We study a class of generalized Besicovitch sets in this thesis. The sets are fractals related to the frequency of the digits in c-ary(c ≥ 2 integer) expansion of the real number x in [0,1] associated to a partition of the natural numbers.Many fractals can be defined in the terms of number theory. For example, the middle-third Cantor set consists of x in [0,1] containing number 0 and 2 in the 3-ary expansion of x. We investegate frcatals associated to the expansion of real numbers. The class of set has been an important study direction in the field of fractal for a long time. For the 2-ary expansion of a real number x ∈ [0,1), a classical result is Borel's theorem on normal numbers in 1909, i.e. all digits appear equally often. Since then, it began to study the set of x that all digits don't appear equally often, that is non-regular numer set. Early in 1934, Besicovitch studied such set in the case of 2-ary expansion, and obtained better result. In 1949, Eggleston generalized the result to the case of more general c-ary expansion.It always has benn hot topic in the field of fractal for the study of Besicovitch-Eggleston sets, and taken many achievements from different ways. Recently Olsen worked over the dimensions of the approximative discrecte Besicovitch-Eggleston sets, and his results provides a natrual discrete analogue of a classical result due to Besicovitch and Eggleston on the Hausdorff dimension of certain sets of non-normal numbers. In addition, Barreira etc. investegate Hausdorff dimension of a large class of sets defineded on real line, which are defined in the terms of the distribution of frequencies of digits in the expansion based on some integers. Their results unify and extend classical work of Borel, Besicovitch, Eggleston and Billingsley in several directions. Their methods are based on recent results concerning the multifractal analysis of dynamical systems andobtain explicit expressions for Hausdorff dimension.We will investegate a class variants of Besicovitch-Eggleston sets. In fact. We will investegate the distribution of frequencies of digits in the expansion corresponding to the positions indexed by a partiton of the natural numbers N. Accurately, let N = UMt be a partition, for any digit j, denote ip/.j(x) by the frequencies of digit j in the positions indexed by the set Mk in the expansion of x. Given P = {pkj}, we study the sets that the frequency ipkj(x) equals to pkj for all or partial (k,j). Their dimensions depend on not only pkj but also the density dk of {Mk} in N(suppose all densities exist).For simplicity, we just consider the sets in the symbolic space. And there is an almost one-to-one correspondence between the symbolic space and the unit interval (considering the c-ary expansion of the real number in [0,1] as a sequence in the symbolic space over a c-letter alphabet). By taking the proper distance in the symbolic space, the correspondence preserves the Hausdorff dimension, thus our results can be also interpreted in the real number case.At first, we deal with modified Besicovitch-Eggleston sets, that is the set of x, whose the frequency tpkj(x) equals to given constant pkj. We firstly work out in the symbolic space with two symbols, then generalize them to general symbolic space. By constructing appropriate measures in the symbolic space, and using Billingsley' theorem and the law of large number, we prove the dimension of set that all frequencies (pkj(x) are given:I c-\dim* E(F) = ~Yldk &*i loScPJy-k-\ j=QAfterward, by a new tool about theory of vectorical multifractal, we provethe dimension of set that partial frequencies (pkj{x) are given: dim// Gv = 1 - YX=i 4 - EjUi d* E"=d Pki loScPkiThis technique is a new one that hasn't been used formerly, and our results are extended by step by step. Therefore, We obtain the exact Hausdorff dimensions of the modified Besicovitch-Eggleston sets in two different methods.Secondly, we study some weighted Besicovitch sets, that consist of points x whose the weighted sum of frequence of digit 1 appearing in the positions indexed by the set Mk in the dyadic representation equuals to some constant p. Their dimensions are given in terms of the constant p, weighted number 8k and the density dk of Mk in N. We also obtain their explicit expressions for Hausdorff dimensions by the same as the approach based on the Billingsley's theorem in symbolic space and our prior result:dimH Ep = □ < ^2 dkh(Ph) ■ Yl S^dkPk = V lWhere h(t) = — tlog2 t - (1 - t) Iog2(l - t). When / is finite, □ stands for max; When I = oo, □ stands for sup.All the results extend the classical work of Borel, Besicovitch, and Eggle-ston etc. And we exploit a new thinking and method of study for Besicovitch-Eggleston sets.