Visible parts and projections of the affine fractal percolation

Abstract

We study the visible parts and orthogonal projections of the affine fractal percolation. We show that for the affine fractal percolation $E$, conditioned on $E \neq \varnothing$, almost surely the dimension of the visible part is equal to $\min\{1, \dim_{\rm H} E\}$ from all directions except for the major contracting direction. The visible part from the major contracting direction is notably different from the other directions, and we provide a formula for its Hausdorff and box dimensions. In addition, we show that almost surely conditioned on $E \neq \varnothing$, $\dim_{\rm H} π(E) =\min\{ 1, \dim_{\rm H} E\}$ for every orthogonal projection $π$ excluding the one onto the major contracting direction, and given $\dim_{\rm H} E >1$, these projections contain intervals.