Assouad type dimensions for self-affine sponges with a weak coordinate ordering condition

Abstract

Recently self-affine sponges have been shown to be interesting counter-examples to several previously open problems. One class of recently studied sponges are Bedford– McMullen sponges with a weak coordinate ordering condition, that is, sponges with several coordinates having the same contraction ratio. The Assouad type dimensions of such sets cannot be calculated using the same formula as the regular Bedford–McMullen sponges. We calculate the Assouad type dimensions for such sponges and the more general Lalley–Gatzouras sponges with a weak coordinate ordering condition, discussing some of their more subtle details along the way.