Weakly separated self-affine carpets

Abstract: In this paper, we study the Hausdorff and box-counting dimensions of diagonally aligned self-affine carpets whose projections onto the $x$ and $y$-axes satisfy the weak separation condition. In particular, we will see that the Hausdorff dimension equals the limit of the Bara\'nski formula, and the box-counting dimension is the limit of the Feng-Wang formula taken over the $n$th level functions. We also prove various equivalent formulas for the box-counting dimension, and we provide an alternative limiting formula for Gatzouras-Lalley carpets. We demonstrate our theorems through two examples that were unobtainable previously, and we calculate their Hausdorff and box-counting dimensions.