The Complex Dimensions of Every Sierpinski Carpet Modification of Dust Type

Abstract

We investigate modified Sierpi\'nski Carpet fractals, constructed by dividing a square into a square $n \times n$ grid, removing a subset of the squares at each step, and then repeating that process for each square remaining in that grid. If enough squares are removed and in the proper places, we get ``Dust Type'' carpets, which have a path-connected complement and are themselves not path-connected. We study these fractals using the Fractal Zeta Functions, first introduced by Michel Lapidus, Goran Radunovi\'c, and Darko \vZubrini\'c in their book \emph{Fractal Zeta Functions and Fractal Drums}, from which we devised an analytical and combinatorial algorithm to compute the complex dimensions of every Sierpi\'nski Carpet modification of Dust Type.