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I really like this one!
Nice formulas, easy implementation and stunning results
Good job and thanks for sharing your paper with us.

I can post some matlab code for 3D and higher, the script itself only outputs an n-dimensional matrix that you'd have to visualize after making some sections if n>3 until you have a 3D matrix then you'll need to plot it as voxels. I could've written my own voxel plotter but i downloaded a free one out of laziness  (PATCH_3DArray)

A Pseudocode for 3D would be like so :
Define precision or grid density g
Define the number of iterations iter
Define k and p (k=3 and p=3 for Menger's sponge for example)

Make 3 arrays xgrid,ygrid,zgrid from a meshgrid (0 to 1 by a step of 1/g) (3D grid coordinates)
Make a 3D matrix of ones of dimensions gxgxg, lets call it Box.

For n=1:iter
    For i=1:number of elements in Box
                Box(i)=Box(i)*sign( -sign((cos(2*pi*(k)^(n-1)*xgrid(i)))+cos(pi/(p))) - sign((cos(2*pi*(k)^(n-1)*ygrid(i)))+cos(pi/(p))) -
sign((cos(2*pi*(k)^(n-1)*zgrid(i)))+cos(pi/(p))) +3)
    End
End

Plot non-zero elements of Box as voxels