1 research outputs found
On a conjecture regarding the upper graph box dimension of bounded subsets of the real line
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]