The q-spread dimension and the maximum diversity of square grid metric spaces

The main aim of this thesis is to compute the growth rate of the q -spread
and the maximum diversity for several square grids at various scales, then
explain some of their characteristic mathematical features.
On one hand, we compute the growth rate of the q -spread of three different
square grids at various scaling. For large n and very small scales
as well as reasonably big scales, the growth rate of the q -spread of these
squares are similar. This occurs because when the scale factor is small, we
can heuristically map the points of the square grid to the solid square, and
we numerically determine that the q -spread is seen to be very close to some
quadratic equation. While, if the scale factor is big, we can approximate the
q -spread of these squares to the q -spread of fnite subset of points in the
middle of these squares and we show that the q -spread of the square grid
over some positive function approaches zero as a scale factor goes to infnity.