Topology of Univoque Sets in Double-Base Expansions

The paper’s statements of Theorem 1.13 (multiplicity) and Theorem 1.15 (topology) are explicit and internally proved via lexicographic characterizations and a careful case analysis by Lemma 1.8; they classify the number of expansions (2, 3, or countably infinite) across cases (i)–(xi) and give when V_Q is closed, when U_Q is closed, Cantor/non-Cantor, and the endpoint criteria for discreteness of V_Q\U_Q . The candidate solution reproduces these outcomes using a compact “first-switch” framework built around the identities π_Q(0α)=π_Q(10^∞)=1/q_1 and π_Q(1µ)=π_Q(01^∞)=1/(q_0(q_1−1)) and a cylinder-interval picture, which align with the paper’s algebra and topology (Remark 1.4(iv) and Theorem 1.18) . The candidate’s switch-based counting dovetails with the paper’s Lemmas 3.2–3.3 describing expansions between a(x) and b(x) or between m(x) and l(x) via 0α↔10^∞ and 1µ↔01^∞ tails, hence yielding 2, 3, or ℵ0 expansions depending on periodicity and shift equalities . One caution: the candidate briefly overstates that V_Q not being a Cantor set “forces” U_Q to be nonclosed in cases (ii),(iii),(vi),(vii), whereas the paper shows subcases in (ii) and (vii) where U_Q can be closed; discreteness and closedness depend on endpoint membership (1/q_1 or 1/(q_0(q_1−1))) as detailed in Theorem 1.15(vii) and Lemma 5.3 . Overall, both reach the same conclusions; the model’s proof strategy is sound and essentially repackages the paper’s ingredients into a “first-switch” principle.