In this talk, a discrete version of the Hausdorff dimension for families of finite point sets is defined.

The following topics are reviewed: some preliminary results deriving from the discrete Hausdorff dimension definition, B.Hunt’s work on the Hausdorff dimension of the graph of Weierstrass functions with random phases, and some relevant theorems from probability theory on the distribution and density of the sum of independent random variables.

In the second section, it is shown that, given dimension \(d\in\mathbb{Z}\) and \(s\in[d-1,d)\), it is possible to construct a function \(f\) such that the family of finite point sets of the form \(P_n=\left\{ (j/q, f(j/q)): j \in \mathbb Z^{d-1} \cap [0,q)^{d-1} \right\}\) with \(n = q^{d-1}\) and \(q\in\mathbb{Z}\) has discrete Hausdorff dimension \(s\). Such a construction involves a mixture of the topics dicussed in the first part.