When does an affine iterated function system preserve an affine subspace for all choices of translation vectors?

The paper’s Theorem 3 proves exactly the two claims at issue: (i) Z_ℓ is a subvariety (hence either all of R^{dN} or of Lebesgue measure zero) and (ii) Z_ℓ = R^{dN} iff every (N−1)-dimensional subspace U is contained in an A-invariant subspace W with dim W ≤ ℓ, where A is the subalgebra generated by A1,…,AN and I. This is established via Lemmas 3.1–3.3, using the reparametrization p_i = (I − A_i)^{-1}v_i and an algebraic linear-dependence criterion for the span of H⋅{p_i − p_N} (H the semigroup generated by the A_i and I) . The candidate solution follows the same core strategy but packages the subvariety claim as a single determinantal condition using a fixed vector-space basis of the unital algebra A and a rank bound on a linear matrix in v. This is equivalent to the paper’s linear-dependence formulation and yields the same dichotomy (full or measure-zero, using the standard lemma on zero sets of nonzero polynomials) . Minor differences: the model uses contraction to justify (I−A_i)^{-1}W ⊆ W via the Neumann series, while the paper notes that only invertibility (1 not an eigenvalue) is needed for Lemma 3.2; and the model omits the explicit dimensional side-condition 1 ≤ N−1 ≤ ℓ < d. Overall, both are correct; the proofs are closely related but presented in different linear-algebraic packages.