New results on embeddings of self-similar sets via renormalization

The uploaded paper explicitly frames the statement “if log α_i is not in the Q-span of {log β_j} for some i, then there is no affine embedding E(X,Y)” as Conjecture 1.1—open in general—and proves it only under additional hypotheses (e.g., algebraic contraction ratios, rank conditions, almost-everywhere parameters) rather than as a general theorem. See the precise formulation of Conjecture 1.1 and its status, plus the new partial results Theorems 1.2 and 1.3, in the paper’s introduction and main results . The broader context (relation to intersections and known special cases) further confirms that the unrestricted statement remains open in the non-homogeneous case . By contrast, the model invokes the claim as an already-established “logarithmic commensurability theorem” for embeddings and concludes E(X,Y)=∅ without any extra assumptions—this is exactly the conjecture and is not known in full generality. The “logarithmic commensurability principle” referenced in the paper concerns invariance of the Q-span of logarithmic scales under choices of generators for a fixed set, not an embedding theorem that would settle the conjecture .