{"id":"4dbbc37f-4f07-459e-a26c-822fb60f7934","arxiv_id":"2501.09582","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For every m at least 3, explicit intervals around k-bonacci numbers q_k are shown to lie in B_m, with positive Hausdorff dimension for the set of points having exactly m base-q expansions.","lead":"The paper constructs explicit intervals of bases q near the k-bonacci numbers where some number x has exactly m base-q expansions and the set of such x has positive Hausdorff dimension. This gives the first known explicit intervals of this kind for every m at least 3.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.5's proof contains a false implication in the q>q_k case; the overlap estimate behind Theorem A needs a corrected derivation.","rationale":"Theorem A's route is: Lemma 2.2 gives τ(π_q(S_{k-1})) > q^{k-4}; Lemma 3.2 preserves this for the trimmed sets P_i and Q_m; Lemma 3.7 shows the convex hulls overlap; Lemma 3.8 gives β_q > 1/8; Theorem F&Y then yields positive Hausdorff dimension. The reader's weakest-assumption pick, Lemma 2.2, is reasonable: it is the main imported quantitative input and both theorems reduce to it. I agree it is worth checking against Sidorov's proof. However, the more immediate, internal obstruction is the q > q_k case of Lemma 3.5: the written proof asserts an implication that is numerically false for k=4, m=1. This is not an attack on the authors; it is a concrete gap in the printed argument. The gap is probably repairable, because a cruder bound on ϵ_q is enough for Lemma 3.8's β_q estimate, so the result itself is not cast into doubt. Still, an accepted manuscript should not contain a false inference at a point that controls the overlap interval. The reader's CONDITIONAL verdict is therefore appropriate, and my read does not move it. I flag Lemma 3.5 as the sharpest place where the current proof, as written, does not close.","tokens_in":27450,"tokens_out":33245,"duration_ms":279954,"concrete_test":"Set k=4, m=1, and q = q_4 + 0.5 q_4^{-15}. Verify that the asserted implication fails: q^{-8}/(q_4-1) ≈ 5.7×10^{-3} is not less than q_4^{-11} ≈ 7.3×10^{-4}. Then re-run the argument of Lemmas 3.7 and 3.8 using only the weaker bound ϵ_q < q^{-k(m+1)}/(q_k-1) instead of Lemma 3.5's target. If β_q > 1/8 still follows, the flaw is a minor expository gap; if not, Theorem A's interval containment is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem A depends on Lemma 3.5, which bounds ϵ_q defined by 1 = π_q((1^{k-1}0)^\\infty) + ϵ_q. In Case 2 (q > q_k) the proof derives ϵ_q ≤ q^{-k(m+1)}/(q_k - 1) and then asserts that this implies ϵ_q < q_k^{-(m+2)k+1}. That implication is false: for k=4, m=1 the displayed upper bound is about q_4^{-8}/(q_4-1) ≈ 5.7×10^{-3}, whereas the claimed target is q_4^{-11} ≈ 7.3×10^{-4}. The hypothesis |q-q_k| < q_k^{-(m+2)k-3} does not repair the algebra, because the displayed bound is already too coarse. Since Lemma 3.5 feeds Lemma 3.7 (the ordering of the convex hulls) and Lemma 3.8 (β_q > 1/8), the application of Theorem F&Y to the sets P_i(q), Q_m(q) is not rigorously justified as written. The lemma appears repairable via a derivative bound, and a weaker bound may suffice, so this is an incomplete-proof concern rather than a disproof; it still requires a revision before Theorem A is fully established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, for q in (1,2), the set U_q^{(m)} of points in I_q=[0,1/(q-1)] having exactly m base-q expansions, and the set B_m of bases for which U_q^{(m)} is nonempty. Theorem A states that for every integer m>=1 and every k>=K_m (with K_m explicit), every q satisfying |q-q_k| < q_k^{-(m+2)k-3} belongs to B_{m+2}, and moreover dim_H(U_q^{(m+2)}) >= 1 - 1024 (m+2)^{20/19} q_k^{4-k} > 0. Here q_k is the k-Bonacci number. This provides the first explicit intervals of bases for which there are points with exactly m base-q expansions when m>=3. Theorem B gives sharper intervals for B_3 using Newhouse's theorem. The proofs combine thickness estimates for the Cantor sets pi_q(S_{k-1}) (Lemma 2.2, imported from Sidorov), a reduction to nonempty intersections of affine images of U_q (Proposition 2.6), and the Falconer-Yavicoli intersection theorem (Theorem F&Y) for Theorem A, respectively Newhouse's theorem (Theorem N) for Theorem B.","tokens_in":27705,"tokens_out":32133,"duration_ms":249856,"significance":"If the proof is completed, this is a solid and useful contribution: it gives the first explicit intervals contained in B_m for all m>=3 and quantitative lower bounds on the Hausdorff dimension of U_q^{(m)} inside those intervals. The constants are explicit and the dependence on m and k is clearly tracked. The paper is careful in stating the external theorems it uses and in identifying the precise quantitative inputs (notably the thickness bound Lemma 2.2). The method is a genuine combination of known intersection theorems with a symbolic construction of the sets P_i(q) and Q_m(q), and the resulting statements are concrete and falsifiable. The main results would remain valuable even if the paper's criticism of a theorem of Sidorov were not correct, because the intervals here are explicit whereas Sidorov's argument (if valid) only gave non-explicit neighbourhoods.","major_comments":[{"comment":"In Lemma 3.5, Case 2, the proof derives the bound ε_q ≤ q^{-k(m+1)}/(q_k−1) and then asserts that this implies ε_q < q_k^{-(m+2)k+1}. This implication is false as written: for k=4, m=1 the displayed upper bound is approximately 5.7×10^{-3}, whereas q_4^{-11} ≈ 7.3×10^{-4}. The origin of the error is that π_q((0^{k−1}1)^∞) equals 1/(q^k−1), not 1/(q_k−1). The same substitution error appears in Eq. (17), where R(Q_m(q)) is written as π_q((1^{k−1}0)^∞) + q^{-km}/(q_k−1), and is propagated in equations (19), (21) and in the proof of Lemma 3.5. Because Lemma 3.5 feeds directly into Lemma 3.7 (the ordering of the convex hulls) and Lemma 3.8 (the bound β_q > 1/8), the proof of Theorem A is not rigorous as written. With the corrected denominator q^k−1 the required inequality is valid, so the error is local and repairable, but it must be fixed and the subsequent estimates re-verified before Theorem A can be accepted.","section":"§3.3, Lemma 3.5 and Eq. (17)"},{"comment":"Several estimates that are load-bearing for Theorem A are asserted with 'it can be checked' or 'routine calculation' rather than proved. These include the passage from (10) to the final bound in Lemma 3.4, the two 'it can be checked' steps in Lemma 3.5, the positivity of the expression in (20), the lower bound in (21), and the bound on the right-hand side of (22) used to justify the choice of c=19/20. Given that one such 'check' in Lemma 3.5 turned out to be false under the displayed formula, the authors should expand these steps into complete, verifiable inequalities or provide a supplementary appendix with the computations. This is not a mere presentation issue because the validity of the dimension estimate in Theorem A depends on these bounds.","section":"§3.4 and §3.5, inequalities (20), (21), (22)"}],"minor_comments":[{"comment":"The statement that Sidorov's theorem in [19] 'contains a mistake which cannot be fixed' is a strong assertion about a published result and is not substantiated anywhere in the paper. Since the main theorems of the paper do not rely on this claim, the authors should either provide a detailed explanation (e.g., in a footnote or appendix) or soften the assertion to avoid making an unproved accusation.","section":"§1, Introduction"},{"comment":"The displayed identity in the proof of Lemma 3.4, Case 1, has an algebra error: the expression for π_q((1^{k−1}0)^∞) − π_{q_k}((1^{k−1}0)^∞) should involve a plus sign between the two bracketed differences, not a minus sign. The subsequent conclusion remains valid because the omitted term has the correct sign in the case considered, but the displayed equality should be corrected.","section":"§3.3, Eq. (9)"},{"comment":"In the sentence 'the factor of 5/4 appears as an easy lower bound for 1/((q_k−1)(q−1))', the word 'lower' should be 'upper': the inequality 1/((q_k−1)(q−1)) < 5/4 is an upper bound.","section":"§4.3, Lemma 4.7"},{"comment":"Expressions such as 'qk−4' are ambiguous in the text; they should be typeset as q^{k−4} to avoid confusion with q_k−4. This applies particularly in Section 3.5 where the thickness lower bound is stated as 'at least qk−4'.","section":"Throughout, notation"}],"recommendation":"major_revision","confidential_remarks":"The central construction and the overall strategy are sound and promising. The main obstacle is the systematic typographical error q_k−1 instead of q^k−1 in Lemma 3.5 and Eq. (17), which invalidates the written proof of Theorem A but appears repairable by a local correction. The authors should also expand the 'routine calculation' steps that are load-bearing. Theorem B relies on Lemmas 4.2 and 4.3 from the authors' own preprint [4]; for journal publication, those results should be either proved in the paper or clearly referenced as accepted/published results. The unsubstantiated critique of Sidorov's theorem should be removed or justified. If these issues are addressed, the paper is likely suitable for publication in a good dynamical-systems or number-theory journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely new: for every m≥3 it produces the first explicit intervals of bases q for which some x has exactly m base-q expansions, with a positive-Hausdorff-dimension guarantee. The engine is a clean application of Falconer–Yavicoli to (m+2)-fold intersections of affine copies of a Cantor set built from Sidorov's S_k sets. That part of the architecture is sound, and the constants are worked out honestly.\n\nThe main soft spot is Lemma 3.5. In Case 2 (q>q_k) the proof derives ϵ_q ≤ q^{-k(m+1)}/(q_k−1) and then asserts this implies ϵ_q < q_k^{-(m+2)k+1}. That implication is false — for k=4, m=1 the upper bound is about 5.7×10^{-3} while the target is about 7.3×10^{-4}. The hypothesis on |q−q_k| does not repair the algebra. Since Lemma 3.5 feeds Lemma 3.7 and Lemma 3.8, Theorem A is not fully justified as written. This looks repairable (a derivative bound should give the needed power), but it is a real gap, not a typo.\n\nTwo smaller issues. Theorem B rests on an unpublished preprint by the same authors ([4]); that citation needs to be available or the lemmas re-proved. And the paper says Sidorov's claimed B_m result has a mistake that cannot be fixed, but gives no proof. That claim is not load-bearing for Theorem A, but if the authors want the positive-measure consequence they should substantiate it.\n\nOverall: the central idea is good, the paper is honest, and the gap is localized. It deserves a serious referee. I'd send it to review with a request to repair Lemma 3.5 and to make the dependency on [4] explicit.","headline":"First explicit intervals in B_m for all m≥3, built on a genuinely new Falconer–Yavicoli argument, but Lemma 3.5 contains a false inequality that must be fixed before Theorem A is fully established.","tokens_in":28252,"tokens_out":3211,"would_cite":false,"duration_ms":27913,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A63","28A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"Explicit q-intervals guarantee numbers with exactly m expansions for every m≥3.","keywords":["base q expansions","non-integer bases","exactly m expansions","Hausdorff dimension","k-Bonacci numbers","thickness of Cantor sets","univoque sets","intersection of Cantor sets"],"falsifier":"For $q = q_{10}$, the tenth special base, enumerate the symbolic dynamics of the maps $f_0(x) = qx$ and $f_1(x) = qx-1$ and count the base-$q$ expansions of points in the switch region; if no point has exactly three expansions, or if the set of such points has Hausdorff dimension zero, then Theorem B's conclusion at this base is false.","tokens_in":27235,"feed_emoji":"🔢","tokens_out":10591,"duration_ms":105051,"temperature":0.7,"pith_summary":"The paper is trying to prove that for every integer $m \\geq 3$ there are intervals of bases $q \\in (1,2)$ in which the set $U_q^{(m)}$ of points with exactly $m$ base-$q$ expansions is nonempty and has positive Hausdorff dimension. Previously this was established only for $m=2$, with a claimed extension to all $m$ resting on a proof the authors believe is flawed. The paper produces the first explicit such intervals, clustered around the special bases $q_k$ near $2$. If the construction is sound, it also settles the open question of whether $B_m = \\{q : U_q^{(m)} \\neq \\emptyset\\}$ has positive Lebesgue measure for every $m \\geq 3$.","feed_headline":"Explicit q-intervals guarantee numbers with exactly m expansions","feed_subtitle":"Near the k-Bonacci bases, such numbers form a set of positive Hausdorff dimension.","key_machinery":"The machinery has three parts. First, the special bases $q_k$, the unique roots in $(1,2)$ of $q^k = q^{k-1} + \\cdots + q + 1$, serve as anchor points: for $q > q_k$, the Cantor set $\\pi_q(S_{k-1})$ of sequences avoiding the words $01^{k-1}$ and $10^{k-1}$ lies inside the set $U_q$ of numbers with a unique expansion, and its gaps are explicitly known intervals. Second, the thickness $\\tau$ of these Cantor sets, the infimum of bridge-to-gap ratios in a gap-removal construction, is bounded below by $q^{k-4}$, and this bound survives the passage to the affine pieces $P_i(q)$ and $Q_m(q)$. Third, an intersection theorem for compact subsets of the line yields positive Hausdorff dimension of a common intersection provided each set has thickness at least $\\tau$, their convex hulls overlap in an interval of relative size at least $1/8$, and a quantitative inequality holds; the paper verifies this inequality with a fixed exponent $c = 19/20$ for all $k$ past a computable threshold. The $m=3$ case instead uses a two-set thickness-product criterion, with strong interleaving proved by explicit points.","core_discovery":"The central claim is that for any $m \\geq 1$, once $k$ is sufficiently large, every $q$ in a tiny window around the $k$-th special base $q_k$ lies in $B_{m+2}$, and $\\dim_H U_q^{(m+2)} \\geq 1 - 1024(m+2)^{20/19} q_k^{4-k} > 0$. The proof replaces the full set of uniquely representable numbers with a Cantor set of sequences that avoid long runs of $0$s and $1$s, cuts this Cantor set into $m+2$ affine pieces whose convex hulls overlap, and invokes an intersection theorem for thick compact subsets of the line to conclude that the common intersection has positive Hausdorff dimension. A separate two-set argument gives a stronger result for $m=3$, producing intervals around $q_k$ for $k \\geq 10$ and a one-sided interval above $q_9$.","pith_inferences":["The same intersection-plus-thickness scheme may generalize to alphabets other than $\\{0,1\\}$ or to other digit sets, provided one has an analogous family of anchor bases and a thickness bound; nothing in the geometric argument appears to depend on the binary alphabet.","The paper's criticism of the earlier claimed general result is logically independent of its own positive results: even if that criticism is wrong, the explicit intervals here are new, and if it is right, they are currently the strongest evidence that every $B_m$ has positive measure.","One could test numerically how far the actual intervals in $B_m$ extend beyond the guaranteed tiny windows, since the construction is deliberately conservative and the true set $B_m$ near $q_k$ may be considerably wider.","The dimension lower bound is probably not optimal; the constants and the choice $c=19/20$ were made for convenience, leaving room for sharper bounds if the underlying thickness estimates are refined."],"forward_implications":["Theorem A gives the first explicit intervals contained in $B_m$ for every $m \\geq 3$.","Since each such interval has positive length, $B_m$ has positive Lebesgue measure for every $m \\geq 3$, answering a question the paper says remains open if an earlier claimed general result is invalid.","The lower bound $\\dim_H U_q^{(m+2)} \\geq 1 - 1024(m+2)^{20/19} q_k^{4-k}$ shows that, along these intervals, the sets of points with exactly $m$ expansions have Hausdorff dimension arbitrarily close to $1$ as $k \\to \\infty$.","Theorem B shows that for $m=3$ the phenomenon already appears for fairly small $k$, not only deep into the sequence of special bases."],"supporting_citations":[{"why":"Supplies the intersection theorem: a family of thick compact subsets of $\\mathbb{R}$ with overlapping convex hulls has intersection of positive Hausdorff dimension; this is the engine of Theorem A.","marker":"[11]"},{"why":"Source of the quantitative thickness bound $\\tau(\\pi_q(S_k)) > q^{k-3}$, re-indexed to $\\tau(\\pi_q(S_{k-1})) > q^{k-4}$, and of the $U_q - U_q$ characterization of $B_2$ that motivates the approach.","marker":"[19]"},{"why":"Gives the gap structure of the Cantor sets $\\pi_q(S_k)$ and the containment $\\pi_q(S_k) \\subset U_q$ for $q > q_k$, used to build the pieces $P_i(q)$ and $Q_m(q)$.","marker":"[12]"},{"why":"Provides the two-set thickness-product criterion used in Theorem B to force a nonempty intersection of $(\\pi_q(S_{k-1})+1)$ and $g_{q,k}(\\pi_q(A_q))$.","marker":"[15]"},{"why":"Earlier work by the same authors supplies the thickness bound $\\tau(\\pi_q(A_q)) > q^{-5}$ and the fixed-expansion construction needed in the proof of Theorem B.","marker":"[4]"}],"fun_headline_variants":["Explicit q-intervals guarantee exactly m expansions","Positive Hausdorff dimension for m-fold base-q numbers","Cantor constructions yield m expansions in explicit q-windows","For every m, tiny windows around q_k have exactly m expansions","Explicit q-windows force exactly m base-q expansions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on an imported bound on how thick a particular Cantor set is—that its gaps are not too large relative to its solid pieces—and the paper does not re-derive this bound.","fun_headline_variants_meta":{"raw":{"variants":["Explicit q-intervals guarantee exactly m expansions","Positive Hausdorff dimension for m-fold base-q numbers","Cantor constructions yield m expansions in explicit q-windows","For every m, tiny windows around q_k have exactly m expansions","Explicit q-windows force exactly m base-q expansions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000912,"raw_usage":{"total_tokens":3858,"prompt_tokens":823,"completion_tokens":3035,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":2954}},"tokens_in":439,"tokens_out":3035,"duration_ms":21356,"temperature":1.0,"reasoning_tokens":2954,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:52:23.634773+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $q = q_{10}$, the tenth special base, enumerate the symbolic dynamics of the maps $f_0(x) = qx$ and $f_1(x) = qx-1$ and count the base-$q$ expansions of points in the switch region; if no point has exactly three expansions, or if the set of such points has Hausdorff dimension zero, then Theorem B's conclusion at this base is false.","supporting_citations":[{"cited_title":"Intersections of thick compact sets in Rd","cited_arxiv_id":null,"evidence_quote":"Supplies the intersection theorem: a family of thick compact subsets of $\\mathbb{R}$ with overlapping convex hulls has intersection of positive Hausdorff dimension; this is the engine of Theorem A."},{"cited_title":"Expansions in non-integer bases: lower, middle and top orders","cited_arxiv_id":null,"evidence_quote":"Source of the quantitative thickness bound $\\tau(\\pi_q(S_k)) > q^{k-3}$, re-indexed to $\\tau(\\pi_q(S_{k-1})) > q^{k-4}$, and of the $U_q - U_q$ characterization of $B_2$ that motivates the approach."},{"cited_title":"Unique representations of real numbers in non-integer bases","cited_arxiv_id":null,"evidence_quote":"Gives the gap structure of the Cantor sets $\\pi_q(S_k)$ and the containment $\\pi_q(S_k) \\subset U_q$ for $q > q_k$, used to build the pieces $P_i(q)$ and $Q_m(q)$."},{"cited_title":"Newhouse","cited_arxiv_id":null,"evidence_quote":"Provides the two-set thickness-product criterion used in Theorem B to force a nonempty intersection of $(\\pi_q(S_{k-1})+1)$ and $g_{q,k}(\\pi_q(A_q))$."},{"cited_title":"On the cardinality and dimension of the slices of Okamoto’s functions","cited_arxiv_id":null,"evidence_quote":"Earlier work by the same authors supplies the thickness bound $\\tau(\\pi_q(A_q)) > q^{-5}$ and the fixed-expansion construction needed in the proof of Theorem B."}],"review_version":1}