INTERSECTIONS OF CANTOR SETS DERIVED FROM COMPLEX RADIX EXPANSIONS

Both the paper and the model obtain density of level sets for the box/ Hausdorff/ packing dimension of C_{n,D} ∩ (C_{n,D}+α) under the same separation hypotheses, by coding α with a Δ-expansion and forcing long runs that alternate “branching” steps (α_j=0 ⇒ multiplicity |D|) with “freezing” steps (α_j = max D − min D ⇒ multiplicity 1). The paper proves the key dimension formula via tile-counting/Moran arguments and then sets β_j ∈ {0, M} using a counting sequence h_j = ⌊jλ⌋ to tune the frequency and achieve any target value, with density obtained by fixing an initial prefix and modifying the tail. The model’s construction is the same in substance, making the disjointness explicit by inserting sufficiently long freezing blocks and then computing the dimension from the average logarithmic branching rate. Any minor differences (e.g., the model’s explicit disjoint-Moran substructure versus the paper’s tile and Moran formalism) are cosmetic.