Rademacher Series And Haus-Dorff Dimension Of Its Level Sets

Weierstrass function is a classical fractal function, which was firstly constructed by Weier-strass in1877, and was improved by Hardy later. In1977, Mandelbrot pointed out that the graph of this function has fractal properties and proved the Hausdorff dimension of the graph is larger than one. He also gave the exact value of the Box dimension of the graph, which deduces the famous conjecture (called dimension conjecture):whether or not the Hausdorff dimension of the graph is equal to the Box dimension of the graph. This dimension conjecture caused a focus on this function. The complexity of the conjecture yields that people consider the deformed Weierstrass function. This thesis studies a mainly deformed Weierstrass function: replacing the cosine function by Rademacher function, which induces a Rademacher series with the form Among the study of level sets Ea={x∈(0,1):∑∞n=1anR(2n-1x)=a}, the typical results are gotten by Beyer, Wu Jun, Wu Min and Xi Li-Feng, where the real sequence{an} satisfies the following condition They gave the Hausdorff dimension of Ea, which is one when the sequence{an} satisfies the condition (H). Beyer added an additional condition that the real sequence{an} belongs to e2, and Wu Jun replaced the additional condition by that the sequence{an-an-1} belongs to e1. Xi Li-Feng removed the condition given by Wu Jun, and Wu Min considered the level sets of complex value Rademacher series.When we study the intersection of several level sets, we can turn it to the study of Rademacher Series of which the coefficient sequence is a sequence of vectors, namely, the real sequence becomes the sequence{an=(a(1)n, a(2)n,..., a(d)n}. Even if each component sequence satisfies the condition (H), the range of a Rademacher series will not fill the whole space, which is a new problem. Hence, if we discuss the intersection of several level sets, we should first determine the range of the Rademacher series.In this paper, we firstly present a detailed discussion of the Rademacher series on the com-plex plane. There are two main difficulties of the study on the range of the Rademacher series with complex coefficient sequence. One is how to ensure the convergence of the series, i.e., how to take a suitable real number x such that the Rademacher series is convergent according to the coefficient sequence. The other one is how to prove the range of a Rademacher series con-tains non-empty interior, i.e., contains interior points. We employ the combinational technique (Lemma3.7) and the Moran iterated functions system to overcome the two difficulties. We get the following denseness theorem.Theorem A Suppose the complex sequence{cn}∞n=1is a non-summable sequence. Then the range of the Rademacher series R({cn}) is dense in the complex plane if and only if the sequence{cn}∞n=1is a linearly non-summable sequence, namely, for any real numbers α,β, that a2+B2(?)0implies that the real sequence{αan+βbn}∞n=1a non-summable sequence.To our surprise, there exists an example (Example5.10), which says that the range of a Rademacher series R({Cn}) is only dense in the complex plane but not equal to the complex plane. In order to discuss whether or not the complex plane is filled with the range, we give the definition of the ratio:t is called a ratio of the complex sequence{cn=an+ibn} if there exists a subsequence{cnk} which is non-summable such that limkâ†'∞ank/bnk=t. By the definition of ratio, we get the theorem of the complex plane filled with the range as following.Theorem B Suppose the complex sequence{cn}∞n=1is a non-summable sequence. If it has two distinct ratios, then the range of the Rademacher series coincides with the complex plane.Under the condition that the range of the Rademacher series is equal to the complex plane, we use a technique named Beurling density to get the following conclusion concerning with level sets.Theorem C Let{cn}∞n=1be a complex sequence. If it is a linearly non-summable se-quence, then, for any ε>0and any complex number c∈C, we have dimH{x∈(0,1) Σ∞n=1cnR{2n-1x)∈B(c, ε)}=1. If it has two distinct ratios, then, for any complex number c∈C, we have dimH Ec=1.Secondly, we study the case that the sequence consists of high-dimensional vectors, where the dimension of vectors is larger than two. In this case, the key Lemma3.7on the complex plane that is used to prove the convergence of a Rademacher series does not work well here, while we employ masks of vectors to overcome this difficulty. We show Theorem A is also true in the high dimension space. Besides, we use the rearrangement of the sequences to overcome the difficulty that the range contains interior points. A vector v∈Rd is called a direction of a vector sequence{an=(an(1),...,an(d))}, if there exists a subsequence{ank} which is non-summable and an integer1≤i≤d such that limkâ†'∞(ank(1)/ank(i),..., ank(d)/ank(i))=v. We get the following theorem, where the definitions of changeable direction and co-direction will be given in Chapter4.Theorem D Let{an}n=1∞be a sequence of Rd. If the sequence of{an}n=1∞has d directions (changeable directions) and those directions (changeable directions) can form a non-singular matrix, then the range of the Rademacher series R({an}) fill the whole space.Theorem E Let{an}n=1∞be a sequence in Rd and a∈Rd be a vector. If the sequence {an} is a linearly non-summable sequence, then for any ε>0, we have dimH{x∈(0,1):∑n≥1anRn(x)∈B(a,ε)}=1. If the sequence has d co-directions and those co-direction can form a non-singular matrix, then dimH Ea=1.Thirdly, the thesis lists some special examples to illustrate the theory of the two above parts. We use the infinitely combinational idea and general Moran set to prove the follow theoremTheorem F Let a sequence of complex numbers{cn=an+ibn} be a linearly non-summable sequence satisfying limnâ†'∞bn/an=0. If there exists a subsequence{cnk}∈l1such that that the integer k>0implies that∑j>k|bnj|>|bnk|,then the range of the Rademacher series coincides with the complex plane itself. In particular, the range induced by the complex sequence{cn=1/n+i/(?)}is equal to the whole complex plane.Finally, the thesis studies Hausdorff dimension of level sets of divergent Rademacher series on the real line. By the Beurling density of the index set and local Holder continuity in the symbolic space, we prove that Hausdorff dimension of level sets of the divergent series is one if the sequence satisfies the condition (H). This conclusion provides a good way to understand the dimensional problem of Weierstrass function.