Some Remarks on the Fractal Structure of Irrigation Balls

The paper is related to a conjecture by Pegon, Santambrogio and Xia concerning the dimension of the boundary of some sets which we are calling “irrigation balls”. We propose a notion of sub-balls and sub-spheres of prescribed radius and we prove that, generically, the only possible Minkowski dimension of sub-spheres is the one expected in the conjecture. At the same time, beside the scale transition properties and the dimension estimates on some significant sets, we propose a third approach to study the fractal regularity which relies on lower oscillation estimates on the landscape function, which turns out to behave as a Weierstrass-type function.