The simplest erasing substitution

The paper’s Proposition 5.7 proves dim_H R^{-1}(y) for rational y equals −log_2 t where t∈(0,1) solves t^2 + t^{1/d} = 1, with d the density of 1’s in the period q of σ(β(y)) (or σ(β′(y))). The candidate’s solution reproduces the same construction: it identifies the fiber as a countable self-similar set indexed by a∈N^m (m=|q|_1), shows the open-set/strong separation, computes the similarity ratios 2^{-(|q|+2∑a_i)}, factors the pressure sum, and derives the identical equation for t. Minor differences are present only in exposition (e.g., parity-based description vs. the paper’s 〈a〉σ(·) formalism), but the arguments match step-by-step, including dyadic edge cases and the irrelevance of finite prefixes. Hence, both are correct and essentially the same proof.