Computing the dimension of self-affine sets is hard because overlaps — regions where different copies of the fractal intersect — create dependencies that defeat standard counting arguments. The usual escape is to assume separation conditions that prevent overlaps, or to prove dimension formulas that hold for “almost all” parameters.
Kirat introduces a perturbation that preserves the overlap structure exactly. Perturbing the translation vectors that define the fractal produces a new set whose overlaps are eventually identical to the original's. Not approximately the same. Not generically the same. Eventually the same — after finitely many iterations of the construction, the overlap patterns coincide.
This is remarkable because perturbations typically destroy overlap structure. Move a translation vector slightly and the overlaps change, which is why “almost all” results work — generic perturbations avoid the problematic overlap configurations. Kirat's perturbations stay within the exact same overlap class, allowing dimension calculations to transfer between the original and perturbed sets.
The method works for integral self-affine sets (integer expanding matrices, integer translation vectors) with irreducible characteristic polynomials, without requiring any separation condition or restriction on the matrix or set parameters. The perturbation establishes the existence of the box dimension and produces an ergodic invariant measure on the toroidal representation with full dimension.
A perturbation that changes the geometry but preserves the combinatorics. The overlaps are the hard part, and they don't move.