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Expansions of real numbers in non-integer bases and charaterisation of Lazy expansion of 1

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Pith's one-line read For non-integer bases, laziness of expansions of 1 is characterized by lexicographic tail inequalities, split at the golden ratio.

desk verdict The paper's main characterization is false: for beta=3/2 the lazy expansion of 1 violates Theorem 6.1(b), and the proof relies on an invalid inequality. read the letter →

arxiv 2412.10378 v1 pith:LGJ77L5R submitted 2024-11-27 math.GM

Vorashil Farzaliyev This is my paper · ORCID
classification math.GM MSC 11A63
keywords beta-expansionsnon-integerbaseslazyexpansiongreedylexicographiccharacterizationgoldenratioexpansionsof1unique
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

Expansions of real numbers in non-integer bases $\beta \in (1,2)$ are usually not unique, so each number has distinguished greedy and lazy expansions. This paper attacks the lazy expansion of the number 1, the first open problem left in the classic characterization of unique expansions. Its main theorem says that a coefficient sequence $(a_i)$ with $1=\sum a_i\beta^{-i}$ is lazy exactly when certain lexicographic inequalities hold between shifted tails $(a_{k+i})$ (or their complements) and $(a_i)$; which inequality is right depends on whether $\beta$ is at most or greater than the golden ratio $\varphi=(1+\sqrt5)/2$. If correct, this gives a complete, testable description of lazy expansions of 1 for every base in $(1,2)$.

What carries the argument

The workhorse object is the lexicographic order on binary coefficient sequences, together with the tail-sum characterization of laziness: an expansion $\sum a_i\beta^{-i}$ is lazy iff $\sum_{i\ge1}(1-a_{k+i})\beta^{-i}<1$ for every $k$ with $a_k=1$. The proof of the new theorem uses this equivalence to convert laziness into a comparison of tails, then builds an inductive upper bound in Lemma 6.1 that forces the forbidden lexicographic cases to make the sum fall below 1; the golden ratio appears as the base above which that upper bound changes sign behaviour.

What would settle it

Take $\beta=3/2$ (which lies between $\varphi$ and 2), compute the lazy expansion of 1 by the lazy algorithm, and test whether $(a_{k+i})>(a_i)$ holds at every $k$ with $a_k=0$; a single counterexample would falsify part (c). A faster check targets the proof: the bound $1-1/(\beta-1)>0$ used in Lemma 6.1 evaluates to $1-2=-1$ at $\beta=3/2$, so the lemma's contradiction step needs a different argument for this base.

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Extended reading notes

Core claim

The central discovery is a lexicographic characterization of laziness for the expansion of 1. Write $1=\sum_{i=1}^\infty a_i\beta^{-i}$ with $a_i\in\{0,1\}$. Theorem 6.1 states: (a) if $(1-a_{k+i})<(a_i)$ whenever $a_k=1$, then $(a_i)$ is lazy; (b) if $\beta\in(1,\varphi]$ and $(a_i)$ is lazy, then the same complement condition holds; (c) if $\beta\in(\varphi,2)$ and $(a_i)$ is lazy, then instead $(a_{k+i})>(a_i)$ whenever $a_k=0$. In words, the position of the shifted coefficient tail relative to the original sequence in the lexicographic order completely determines whether the expansion is the smallest one, and the golden ratio marks where the criterion switches from complements to direct shifts.

Load-bearing premise

The forward direction of the characterization depends on the inequality $1 - 1/(\beta-1) > 0$ for every $\beta\in(\varphi,2)$; if that bound fails for some base in the interval, the contradiction in Lemma 6.1 does not go through.

Editorial extensions

If this is right

  • For every base $\beta\in(1,2)$, a candidate sequence for 1 can be certified as lazy or not by comparing shifted tails lexicographically with the original sequence.
  • The base interval splits at $\varphi=(1+\sqrt5)/2$: at or below $\varphi$ laziness is captured by the complement condition on positions where $a_k=1$; above $\varphi$ by the direct condition on positions where $a_k=0$.
  • The theorem resolves the first open problem of the cited 1990 paper on unique expansions, if its inequalities hold.
  • It also gives a symbolic, order-theoretic test that does not require computing the whole expansion numerically.

Reading between the lines

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

  • Editorial inference: the same lexicographic pair of conditions may also characterize lazy expansions of arbitrary $x\in[0,1/(\beta-1)]$, after replacing the target 1 in the tail inequalities by $x$.
  • Editorial inference: because laziness is defined as the lexicographically smallest expansion, the theorem suggests a prefix-check algorithm that could certify laziness from finitely many tail comparisons once a quantitative bound on the switching position is known.
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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

4 major / 6 minor

Summary. The paper studies beta-expansions of real numbers in non-integer bases, develops algorithms for generating such expansions, and states a characterization of the lazy expansion of 1 in Theorem 6.1, which it claims solves the first open problem in the cited Erdős-Komornik paper. The manuscript also contains a chapter on replacing words in beta-expansions and a theorem on unique expansions of 1 (Theorem 5.1). The central claims are supported by proofs of several lemmas, but the main characterization theorem is false, and the supporting arguments contain invalid inferences.

Significance. If Theorem 6.1 were correct, it would resolve a known open problem in beta-expansions and would be a noteworthy contribution. The paper also provides a readable exposition of standard material on greedy and lazy expansions and illustrates non-uniqueness through word replacement. These expository parts are not original research contributions, and the core result fails: Theorem 6.1(b) is directly contradicted by a concrete example, and the proofs of Lemma 6.1 and Lemma 6.2 contain load-bearing errors. The paper's main claim is therefore not established.

major comments (4)
  1. [Theorem 6.1(b)] This statement is false. For beta = 3/2, which lies in (1, phi], Definition 4.2 gives the lazy expansion of 1 as a = (0,1,0,...): at i=1, the tail sum sum_{j>1} beta^{-j} = 1/(beta(beta-1)) = 4/3 >= 1, so a_1=0; at i=2, the tail sum is 8/9 < 1, so a_2=1; at i=3, a_1/beta + a_2/beta^2 + sum_{j>3} beta^{-j} = 4/9 + 16/27 > 1, so a_3=0. Taking k=2 (with a_2=1), the sequence (1-a_{k+i}) begins with 1-a_3 = 1, whereas (a_i) begins with a_1 = 0. Hence (1-a_{k+i}) is lexicographically greater than (a_i), contradicting the claimed inequality. The forward direction of Theorem 6.1(b) is therefore false.
  2. [Lemma 6.1, proof] The proof of the forward direction asserts that 1 - (B) > 0 because, for beta in ((1+sqrt5)/2, 2), one has 1 - 1/(beta-1) > 0. This inequality is false: beta-1 lies between (sqrt5-1)/2 and 1, so 1/(beta-1) > 1, making 1 - 1/(beta-1) negative. The derived upper bound (B) < 1, and the contradiction x < 1, do not follow. In addition, the argument only treats the case k=1, since it uses 'a_1 = 1'; even a corrected inequality would not establish the claim for every k with a_k=1.
  3. [Theorem 5.1(b)] The converse direction of Theorem 5.1(b) is unsupported. Lemma 5.4(b) shows that the condition (1-a_{k+i}) < (a_i) whenever a_k=1 implies that (a_i) is lazy, not that the expansion is unique. The statement is in fact false: for beta = phi = (1+sqrt5)/2, the lazy expansion of 1 is (0,1,1,1,...); this sequence satisfies (1-a_{k+i}) < (a_i) for every k with a_k=1, yet the greedy expansion (1,1,0,0,...) is a different expansion of 1. Thus uniqueness does not follow from the stated condition.
  4. [Lemma 6.2 and Theorem 6.1(c)] Lemma 6.2 does not establish Theorem 6.1(c). The proof considers only k=1, showing that (a_{1+i}) cannot equal or be lexicographically less than (a_i); it does not show that (a_{k+i}) > (a_i) for arbitrary k with a_k=0. The lemma statement also has a notational error, using '(a_{k+1})' where the context requires '(a_{k+i})'. Consequently, Theorem 6.1(c) does not follow from the cited lemmas.
minor comments (6)
  1. [Title/Abstract] The title contains the typo 'charaterisation' and the abstract uses 'nominator' for 'numerator'; these should be corrected.
  2. [Introduction, p. 6] The introduction states that beta-expansions are for beta in (0,1), but the entire paper works with beta in (1,2); this is a significant notational inconsistency that should be fixed.
  3. [Chapter 5] The proofs refer to 'Lemma 10.1', 'Lemma 10.3', and 'Lemma 10.4', but the numbering in the manuscript is Lemma 5.1, Lemma 5.3, and Lemma 5.4; the cross-references are inconsistent.
  4. [Lemma 6.1 statement] The lemma statement writes '(1-a_{k+1})' but the proof and Theorem 6.1 require '(1-a_{k+i})'; this notational slip should be corrected.
  5. [Section 2.2, words of length 4] For the word 0001, the polynomial is miscomputed: 1/beta^4 = 1/beta + 1/beta^2 + 1/beta^3 gives beta^3 + beta^2 + beta - 1 = 0, not beta^3 - beta^2 - beta - 1 = 0 as written.
  6. [Lemma 2.1 proof] The solution of beta^2 - beta - 1 = 0 is beta = (1 + sqrt5)/2 or beta = (1 - sqrt5)/2, not 'beta = +/- (1+sqrt5)/2' as written; the negative root is not of the stated form.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained or uses the external Erdős–Komornik characterization; the paper's serious defects are correctness errors, not circular reasoning.

full rationale

I walked the derivation chain from the algorithms and definitions through Theorem 6.1. Definitions 4.2/4.3 define lazy/greedy as lexicographic extrema; Lemma 5.1 (quoted from Erdős–Komornik) gives numerical sum characterizations; Lemmas 5.2–5.4 are derived internally from that lemma. Theorem 6.1(a) is literally Lemma 5.4(b) instantiated at x=1, so it is a corollary or restatement rather than an independent new result, but it is not circular because Lemma 5.4(b) is proved earlier in the paper using Lemma 5.3, and the lexicographic condition is not the definition of laziness. Theorem 6.1(b) and (c) are assembled from Lemmas 6.1 and 6.2, which are attempted necessity proofs; they do not fit any parameter or presuppose the conclusion. The paper contains a serious mathematical error—Lemma 6.1 uses the false bound '1 - 1/(β-1) > 0' for β in (φ,2), and Theorem 6.1(b) is contradicted by β=3/2, whose lazy expansion of 1 begins 0,1,0,1,...—but false inference is a correctness defect, not a circular reduction. There are no fitted inputs called predictions, no load-bearing self-citations, and no uniqueness theorem imported from the present authors. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted. The paper introduces maps T and R as algorithmic tools, but these are constructions rather than invented physical or mathematical entities. The main unstated assumptions are standard beta-expansion existence results and the correctness of the paper's stated Lemma 5.1(b) and Theorem 5.1(b).

assumptions (3)
  • domain assumption Greedy and lazy beta-expansions exist for all x in [0, 1/(beta-1)] for beta in (1,2).
    Used throughout Chapters 1 and 4 without proof; standard in beta-expansion theory.
  • ad hoc to paper Lemma 5.1(b): A sequence is lazy iff sum (1 - a_{k+i}) beta^{-i} < 1 whenever a_k = 1.
    This lemma is used in Lemma 6.1 and Theorem 6.1, but its proof depends on the algorithms from Chapter 1 and the lexicographic definitions. It is a central tool.
  • ad hoc to paper Theorem 5.1(b) as stated, claiming uniqueness from the single condition (1 - a_{k+i}) < (a_i) whenever a_k = 1.
    The theorem statement omits the greedy-side condition that appears in the proof of Lemma 5.2(b). Lemma 5.4(b) only establishes laziness, so uniqueness does not follow. This misstatement is used to frame the lazy characterization.

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Pith. "Pith review of Expansions of real numbers in non-integer bases and charaterisation of Lazy expansion of 1." pith.science (2026). https://pith.science/paper/LGJ77L5R

@misc{pith2026241210378,
  author       = {Pith},
  title        = {Pith review of: Expansions of real numbers in non-integer bases and charaterisation of Lazy expansion of 1},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LGJ77L5R}},
  note         = {Machine review of arXiv:2412.10378}
}
abstract

In this paper, our main focus is expressing real numbers on the non-integer bases. We denote those bases as $\beta$'s, which is also a real number and $\beta \in (1,2)$. This project has 3 main parts. The study of expansions of real numbers in such bases and algorithms for generating them will contribute to the first part of the paper. In this part, firstly, we will define those expansions as the sums of fractions with $1$'s or $0$'s in the nominator and powers of $\beta$ in the denominator. Then we will focus on the sequences of $1$'s and $0$'s generated by the nominators of in the sums we mentioned above. Such sequences will be called \textit{coefficient sequences} throughout the paper. In the second half, we will study the results in the first chapter of \cite{erdos1990characterization}, namely the greedy and lazy $\beta $-expansions . The last part of the paper will be on the characterisation of lazy expansion of 1, which was the first open question at the end of \textit{Erdos and Komornik}. I still don't know if that problem has been solved already. However, the solution that was presented here is the original work of mine.

Figures

Figures reproduced from arXiv: 2412.10378 by the authors.

Figure 1.1
Figure 1.1. Diagram of 𝑇 (𝑥) 13 [PITH_FULL_IMAGE:figures/full_fig_p014_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Diagram of 𝑅(𝑥) 16 [PITH_FULL_IMAGE:figures/full_fig_p017_1_2.png] view at source ↗
Figure 2.1
Figure 2.1. Diagram of 𝑇 (𝑥), and 𝑅(𝑥) We divide the region [0, 1 𝛽−1 ] into 3 non-overlapping regions, as follows [0, 1 𝛽 − 1 ] = [0, 1 𝛽 ) ∪ [ 1 𝛽 , 1 𝛽(𝛽 − 1) ] ∪ ( 1 𝛽(𝛽 − 1) , 1 𝛽 − 1 ] On the first region [0, 1 𝛽 ), 𝑇 (𝑖) (𝑥) and 𝑅(𝑖) (𝑥) overlap also recall from the Section 1.2 and 1.3 𝑇 (𝑖) (𝑥) = 𝛽𝑇 (𝑖−1)(𝑥) − 1 for all 𝑖 ∈ ℕ 𝑅 (𝑖) (𝑥) = 𝛽𝑅(𝑖−1)(𝑥) for all 𝑖 ∈ ℕ Hence, 𝑎𝑖 = 𝑏𝑖 = 0 if we lay on this region at some step i… view at source ↗

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Works this paper leans on

1 extracted references · 1 canonical work pages

  1. [1]

    Characterization of the unique expansions1 = ∑∞ 𝑖=1𝑞−𝑛𝑖 and related problems

    [1] Erdös, Pál, Joó, István, and Komornik, Vilmos. "Characterization of the unique expansions1 = ∑∞ 𝑖=1𝑞−𝑛𝑖 and related problems."Bulletin de la Société Mathématique de France, vol. 118, no. 3, 1990, pp. 377–390. 49

    work page 1990

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