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.