关于实直线有界子集的上图盒维数的一个猜想
On a conjecture regarding the upper graph box dimension of bounded subsets of the real line
摘要：
设X \subset R是一个有界集;我们引入一个计算X的上图盒维数的公式（即定义在X上的所有一致连续函数上的图的上盒维数的上确界）。我们通过计算某些集合的上图盒维数和给出一个“一行”证明来证明这个公式的强度，以代替[1]中给出的证明，证明如果X有n个孤立点，则它的上图盒维数等于上盒维数加1。此外，我们构造了一个集合X的无限多个孤立点，有上盒维数a采取值从0到1，而他们的图盒维数采取任何值在[max{2a，1}，a + 1]，回答这种方式，否定了一个猜想在[1]。
英文摘要：
Let X \subset R be a bounded set; we introduce a formula that calculates the upper graph box dimension of X (i.e.the supremum of the upper box dimension of the graph over all uniformly continuous functions defined on X). We demonstrate the strength of the formula by calculating the upper graph box dimension for some sets and by giving an "one line" proof, alternative to the one given in [1], of the fact that if X has finitely many isolated points then its upper graph box dimension is equal to the upper box dimension plus one. Furthermore we construct a collection of sets X with infinitely many isolated points, having upper box dimension a taking values from zero to one while their graph box dimension takes any value in [max{2a,1},a + 1], answering this way, negatively to a conjecture posed in [1].
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