On a conjecture regarding the upper graph box dimension of bounded subsets of the real line

TL;DR

This paper introduces a formula for calculating the upper graph box dimension of bounded sets in the real line, demonstrating its effectiveness through examples and disproving a previous conjecture with new set constructions.

Contribution

It provides a new formula for the upper graph box dimension and constructs sets that counter a prior conjecture about its bounds.

Findings

Calculated upper graph box dimension for various sets

Provided a simplified proof for sets with finitely many isolated points

Counterexample sets show the conjecture does not hold in general

Abstract

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].