A new fractal dimension for discrete structures: Frostman meets Moran

This paper introduces and systematically investigates the Frostman dimension—a novel fractal dimension defined for sequences of discrete metric spaces, inspired by the energy form of Frostman’s lemma. While Frostman’s lemma has long been a cornerstone in estimating the Hausdorff dimension of continuous fractals, its adaptation to discrete settings remains largely unexplored. We focus on discrete Moran sets on the integers—a class of fractal-like sets constructed recursively using integer-valued scaling parameters { c k } and digit sets { A k } . By defining the Frostman dimension for the sequence of finite approximations { Y k } of such a set, we establish a surprising result: this dimension coincides with the upper asymptotic density of the scaling process, explicitly given by Dim ( { Y k } ) = lim ¯ k → ∞ log ( n 1 n 2 ⋯ n k ) log ( c 1 c 2 ⋯ c k ) , where n k = ♯ A k . This is in stark contrast to the classical Hausdorff dimension of Moran fractals in R d , which typically involves a lower limit and contraction ratios r k . We further provide an explicit example where the Frostman dimension diverges from the natural lower limit, underscoring its unique nature. A key contribution is the development of a matrix-based estimation technique to control the energy sums ∑ | x − y | − s even when the strong separation condition fails. Our work thus bridges geometric measure theory with discrete fractal analysis by introducing the Frostman dimension to discrete Moran sets, offering a new tool for classifying the complexity of integer-based structures and opening avenues for further research into dimension theory on discrete spaces.