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On finitely many base $q$ expansions

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Pith's one-line read Explicit q-intervals guarantee numbers with exactly m expansions for every m≥3.

desk verdict 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. read the letter →

arxiv 2501.09582 v1 pith:VBU4ASF3 submitted 2025-01-16 math.DS math.NT

Simon Baker , George Bender This is my paper · ORCID
classification math.DSmath.NT MSC 11A6328A80
keywords baseqexpansionsnon-integerbasesexactlymHausdorffdimensionk-BonaccinumbersthicknessofCantorsetsunivoqueintersection
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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 $m \geq 3$.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $q_9$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [§3.3, Lemma 3.5 and Eq. (17)] 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.
  2. [§3.4 and §3.5, inequalities (20), (21), (22)] 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.
minor comments (4)
  1. [§1, Introduction] 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.
  2. [§3.3, Eq. (9)] 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.
  3. [§4.3, Lemma 4.7] 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.
  4. [Throughout, notation] 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'.

Circularity Check

1 steps flagged · score 1.0 of 10

No definitional circularity; Theorem A is self-contained and Theorem B carries only a minor self-citation provenance burden.

  1. self citation load bearing [Section 4.1, Eq. (25), Lemmas 4.2 and 4.3, applied in Section 4.4]
    "it has been shown in previous work by the authors [4, Proposition 3.13] that the set Aq we will construct satisfies τ (πq(Aq)) > q−5, (25) when q > q9."

    Theorem B's Newhouse intersection argument needs the quantitative thickness bound τ(π_q(A_q)) > q^{-5} and the containment π_q(A_q) ⊂ U_q ∩ (U_q+1), both imported from the same authors' earlier preprint [4, Props. 3.10 and 3.13]. The present paper does not re-derive these facts, so the proof of Theorem B is not self-contained and relies on a load-bearing self-citation. It is nevertheless not a reduction by definition: [4] is an independent study of slices of Okamoto's functions, and the target statement q ∈ B_3 is not an input to [4]. Thus this is a provenance concern rather than a forced or equivalent derivation.

full rationale

Theorem A is derived from external, non-circular inputs: the Falconer-Yavicoli intersection theorem, Sidorov's thickness estimate for π_q(S_{k-1}), and structural lemmas about Cantor sets of unique expansions. The explicit intervals around q_k are not chosen to force the conclusion; their radii are obtained from explicit eps_q estimates and thickness bounds. No fitted parameter is relabelled as a prediction, and the target sets B_m are not used in their own proof. The only self-citation of note is in Theorem B, where the A_q construction and its thickness are quoted from the authors' prior work; this is load-bearing for Theorem B but independent of Theorem A and does not make the overall derivation equivalent to its inputs. The possible issue in Lemma 3.5 flagged by a skeptical reading is a correctness or repair concern, not a circularity concern.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central claim rests on established geometric measure theory theorems, on prior structure and thickness results for unique-expansion Cantor sets, and, for Theorem B, on two lemmas from the authors' own unpublished preprint. The only free parameter is the proof constant c, which is chosen for convenience rather than fitted to data.

free parameters (1)
  • c = 19/20
    Hand-chosen exponent in the Falconer-Yavicoli application in the proof of Theorem A, Section 3.5. The authors note that numerical testing showed it is reasonable and that they do not believe another value of c strengthens Theorem A.
assumptions (5)
  • standard math Falconer-Yavicoli Theorem F&Y, special case of [11, Theorem 6]
    Used as a black box to convert thickness, convex hull overlap, and a quantitative scale condition into a positive Hausdorff dimension lower bound for the intersection of m+2 compact sets. This underpins Theorem A.
  • standard math Newhouse thickness theorem, Theorem N from [15]
    Used for the two-set intersection in Theorem B: interleaving plus thickness product at least 1 implies nonempty intersection.
  • domain assumption Glendinning-Sidorov structural lemma, Lemma 2.1 from [12, Lemma 4]
    Gives pi_q(S_k) subset U_q and describes all gaps of these Cantor sets. This is used throughout the construction of P_i, Q_m, and in verifying thickness preservation.
  • domain assumption Sidorov thickness estimate, Lemma 2.2 from [19, Theorem 4.4]
    Gives tau(pi_q(S_{k-1})) > q^{k-4}, the key quantitative input for both the Falconer-Yavicoli and Newhouse applications in Theorems A and B.
  • domain assumption Baker-Bender preprint lemmas, Lemmas 4.2 and 4.3 from [4]
    Theorem B imports from the authors' own arXiv preprint the assertions pi_q(A_q) subset U_q intersect (U_q+1) and tau(pi_q(A_q)) > q^{-5}. These are not re-proved in this paper.
invented entities (1)
  • A_q
    purpose: A set of digit sequences whose projection lies in U_q intersect (U_q+1), used to prove Theorem B via a two-set Newhouse intersection.
    Auxiliary set defined via a fixed expansion of 1; existence of such expansions is proved in Lemma 4.1, and the U_q containment is imported from [4]. It is not a postulated physical entity and does not need external detection.

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Pith. "Pith review of On finitely many base $q$ expansions." pith.science (2026). https://pith.science/paper/VBU4ASF3

@misc{pith2026250109582,
  author       = {Pith},
  title        = {Pith review of: On finitely many base $q$ expansions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBU4ASF3}},
  note         = {Machine review of arXiv:2501.09582}
}
abstract

Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any $q$ in one of these intervals, the set of points with exactly $m$ base $q$ expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of $\mathbb{R}^d$ has positive Hausdorff dimension under certain conditions.

Figures

Figures reproduced from arXiv: 2501.09582 by the authors.

Figure 2
Figure 2. The process of taking subsets of affine images of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The relative structure of the convex hulls of the sets [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · cited by 1 Pith paper

  1. [4]

    On the cardinality and dimension of the slices of Okamoto’s functions

    Simon Baker and George Bender. On the cardinality and dimension of the slices of Okamoto’s functions. arXiv e-prints , pages arXiv–2310, 2023

    work page 2023

  2. [1]

    Entropy, topological transitivity, and dimensional properties of unique q-expansions

    Rafael Alcaraz Barrera, Simon Baker, and Derong Kong. Entropy, topological transitivity, and dimensional properties of unique q-expansions. Trans. Amer. Math. Soc. , 371(5):3209–3258, 2019

    work page 2019

  3. [2]

    Generalized golden ratios over integer alphabets

    Simon Baker. Generalized golden ratios over integer alphabets. Integers, 14:Paper No. A15, 28, 2014

    work page 2014

  4. [3]

    On small bases which admit countably many expansions

    Simon Baker. On small bases which admit countably many expansions. J. Number Theory , 147:515–532, 2015

    work page 2015

  5. [5]

    Expansions in non-integer bases: lower order revisited

    Simon Baker and Nikita Sidorov. Expansions in non-integer bases: lower order revisited. Integers, 14:Paper No. A57, 15, 2014

    work page 2014

  6. [6]

    Metric results for numbers with multiple q-expansions

    Simon Baker and Yuru Zou. Metric results for numbers with multiple q-expansions. J. Fractal Geom. , 10(3-4):243–266, 2023

    work page 2023

  7. [7]

    On the structure of univoque numbers

    Zolt´ an Dar´ oczy and Imre K´ atai. On the structure of univoque numbers. Publ. Math. Debrecen , 46(3- 4):385–408, 1995

    work page 1995

  8. [8]

    On the uniqueness of the expansions 1 =P q−ni

    P´ al Erd¨ os, Mikl´ os Horv´ ath, and Istv´ an Jo´ o. On the uniqueness of the expansions 1 =P q−ni . Acta Math. Hungar., 58(3-4):333–342, 1991

    work page 1991

Show all 21 references
  1. [9]

    On the number of expansions 1 =P q−ni

    P´ al Erd¨ os and Istv´ an Jo´ o. On the number of expansions 1 =P q−ni . II. Ann. Univ. Sci. Budapest. E¨ otv¨ os Sect. Math., 36:229–233, 1993

    1993

  2. [10]

    Characterization of the unique expansions 1 =P∞ i=1 q−ni and related problems

    P´ al Erd¨ os, Vilmos Komornik, and Istv´ an Jo´ o. Characterization of the unique expansions 1 =P∞ i=1 q−ni and related problems. Bulletin de la Soci´ et´ e Math´ ematique de France, 118(3):377–390, 1990

    1990

  3. [11]

    Intersections of thick compact sets in Rd

    Kenneth Falconer and Alexia Yavicoli. Intersections of thick compact sets in Rd. Math. Z. , 301(3):2291– 2315, 2022. 22

    2022

  4. [12]

    Unique representations of real numbers in non-integer bases

    Paul Glendinning and Nikita Sidorov. Unique representations of real numbers in non-integer bases. Math. Res. Lett., 8(4):535–543, 2001

    2001

  5. [13]

    Hausdorff dimension of univoque sets and devil’s staircase

    Vilmos Komornik, Derong Kong, and Wenxia Li. Hausdorff dimension of univoque sets and devil’s staircase. Adv. Math., 305:165–196, 2017

    2017

  6. [14]

    Unique developments in non-integer bases

    Vilmos Komornik and Paola Loreti. Unique developments in non-integer bases. Amer. Math. Monthly , 105(7):636–639, 1998

    1998

  7. [15]

    Newhouse

    Sheldon E. Newhouse. Nondensity of axiom A(a) on S2. In Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) , volume XIV-XVI of Proc. Sympos. Pure Math. , pages 191–

    1968

  8. [16]

    On the β-expansions of real numbers

    William Parry. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar , 11:401–416, 1960

    1960

  9. [17]

    Representations for real numbers and their ergodic properties.Acta Math

    Alfr´ ed R´ enyi. Representations for real numbers and their ergodic properties.Acta Math. Acad. Sci. Hungar, 8(3-4):477–493, 1957

    1957

  10. [18]

    Almost every number has a continuum of β-expansions

    Nikita Sidorov. Almost every number has a continuum of β-expansions. Amer. Math. Monthly, 110(9):838– 842, 2003

    2003

  11. [19]

    Expansions in non-integer bases: lower, middle and top orders

    Nikita Sidorov. Expansions in non-integer bases: lower, middle and top orders. J. Number Theory , 129(4):741–754, 2009

    2009

  12. [20]

    Ergodic properties of the Erd¨ os measure, the entropy of the golden shift, and related problems

    Nikita Sidorov and Anatoly Vershik. Ergodic properties of the Erd¨ os measure, the entropy of the golden shift, and related problems. Monatsh. Math. , 126(3):215–261, 1998. 23

    1998

  13. [202]

    Amer. Math. Soc., Providence, RI, 1970

    1970

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