On the horizon problem over the product of self-similar fractals

Abstract

In this article, we first investigate the box dimension results for the graphs of continuous functions defined on the product of two self-similar fractals ({\mathfrak{F}}s), where the fractal {\mathfrak{F}} satisfies the open set condition. Following this, we manifest the existence of a prevalent surface, defined by a continuous function over the product of two {\mathfrak{F}}s, that satisfies the horizon property (associated with the box dimension), i.e., the box dimension of the prevalent surface is greater than that of its horizon up to a constant η, where η is the box dimension of the self-similar fractal {\mathfrak{F}}.