EXACT OVERLAPS ON THE PROJECTIONS OF THE FOUR CORNER CANTOR SET

The paper proves that Ct has an exact overlap iff t = p/q in lowest terms with p,q not in Γ = {(2k−1)·2^{2ℓ−1}} (i.e., both v2(p), v2(q) are even), via: (i) no overlaps for irrational t (Proposition 2.1), (ii) Kenyon’s injectivity criterion yielding the divisibility condition (Proposition 2.3), and (iii) a linear-algebraic rank analysis (Lemma 2.4, Theorem 2.5) implying the Γ-condition; combined in Theorem 1.1 and extended to t>0 . The candidate solution reaches the same characterization and gives a clean “only-if” 2-adic valuation argument; its converse direction informally appeals to Kenyon’s injectivity and a matrix analysis akin to the paper’s. One step (a claimed gcd formulation with x^q±1) is not supported by Proposition 2.3 as stated in the paper, but this does not affect the final conclusion.