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Some Bounds Related to the $2$-adic Littlewood Conjecture

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Pith's one-line read This paper proves concrete lower bounds for the 2-adic Littlewood conjecture: every irrational $\alpha$ has some iterate $2^k\alpha$ with a continued-fraction partial quotient at least 15, and some iterate whose eventual partial quotients…

desk verdict The B-variant lower bound is a genuine new idea but the proof has a load-bearing gap; the M-bound is already superseded. read the letter →

arxiv 2506.04110 v2 pith:KCLAXVO7 submitted 2025-06-04 math.NT

Dinis Vitorino , Ingrid Vukusic This is my paper · ORCID
classification math.NT MSC 11A5510A3011J13
keywords 2-adicLittlewoodconjecturecontinuedfractionspartialquotientscontinued-fractiondoublingalgorithmquadraticirrationalsbadlyapproximablenumberslimsupof
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 2-adic Littlewood conjecture asks whether, for every irrational real number $\alpha$, iterated doubling $2^k\alpha$ eventually produces unbounded continued-fraction partial quotients. This paper does not settle that conjecture, but it shrinks the room for a counterexample: the first theorem says that if every double of $\alpha$ had partial quotients below 15, that would already contradict what is known, so any counterexample must tolerate a digit of size at least 15. The second theorem attacks the weaker 'eventual' statistic $B(\alpha)=\limsup_{n\to\infty} a_n(\alpha)$ and proves that some double of every irrational has eventual partial quotients reaching at least 5. These are unconditional constraints on the two outstanding conjectures, and the second is the paper's main new construction.

What carries the argument

The engine is the classical multiplication-by-2 algorithm for continued fractions, derived from the identities $2\cdot[a;2m,b,\theta]=[2a;m,2b,\theta/2]$ and $2\cdot[a;2m+1,\theta]=[2a;m,1,1,(\theta-1)/2]$. The algorithm reads a window $(a,b,c)$ of digits of $\alpha$, outputs either $a/2,2b$ or $(a-1)/2,1,1$, then slides right, and a cleanup step merges zero digits. For the B-bound the key fact is that the three operations $2x$, $x/2$, and $(x+1)/2$ visit the same window in three distinct ways, never colliding, which forces digit patterns in one operation to constrain the others. The longest-chain construction uses roots of $x^2-mx-2$, whose doubling orbit is self-equivalent, giving $\alpha\sim\alpha/2\sim(\alpha+1)/2$ and hence arbitrarily long finite chains $\beta,2\beta,\ldots,2^K\beta$ with the same periodic tail.

What would settle it

Re-run the prefix-exclusion computation for C=14 with an independent implementation; if any prefix survives to the claimed stopping point, or if a calculated output digit is ever smaller than the digit forced by Algorithm 1, Theorem 2's bound of 15 is wrong. For the B-bound, enumerate by computer all periodic digit tails that keep the three doubling-related algorithms below 4; any tail not equivalent to $[3;1,1]$ that survives would falsify Lemma 16 and Theorem 6.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the two forms of the 2-adic Littlewood conjecture admit finite local obstructions that can be made explicit. Theorem 2 states $\sup_{k\ge 0} M(2^k\alpha) \ge 15$ for every irrational $\alpha$, proved by a computer-assisted exhaustion of all possible prefixes of continued fractions whose digits are at most 14; consequently $q\cdot|q|_2\cdot\|q\alpha\|<1/15$ for every $\alpha$ (Corollary 1). Theorem 6 states $\sup_{k\ge 0} B(2^k\alpha) \ge 5$, and the proof shows the mechanism: under the multiplication-by-2 algorithm, the digit patterns that could evade a bound of 4 collapse to the periodic tail $[3;1,1]$, and the only alternative produces a digit 8. The paper also constructs, for every odd $m\ge 3$ and every $K$, quadratic irrationals $\alpha_{m,K}$ with $B(2^k\alpha_{m,K})=m$ for all $0\le k\le K$, showing that such local obstructions can persist for arbitrarily long finite doubling chains.

Load-bearing premise

The load-bearing premise is that the two-hour computer search that excluded all prefixes for C=14 is free of bugs, and that the window-by-window case analysis behind the B-bound is exhaustive.

Editorial extensions

If this is right

  • Any counterexample to 2LC must satisfy $\sup_{k\ge 0} M(2^k\alpha)\ge 15$, improving the previously known lower bound of 8.
  • Any counterexample to the B-variant must satisfy $\sup_{k\ge 0} B(2^k\alpha)\ge 5$, ruling out every potential counterexample whose eventual partial quotients stay at 4 or below.
  • For every irrational $\alpha$ there is an integer $q$ with $q\,|q|_2\,\|q\alpha\|<1/15$, an explicit strengthening of the earlier constant $1/9$.
  • The equivalence between finite-horizon and global counterexamples holds for $M$ but not for $B$, so the B-bound requires different methods than computer prefix search.

Reading between the lines

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

  • One step beyond the paper: the same transducer view suggests compiling the multiplication-by-2 algorithm into a finite automaton over parity classes of convergent denominators; searching for an infinite accepting path with all outputs below 5 would directly test whether the B-bound of 5 is tight.
  • Not in the paper: a limit of the quadratic-irrational chains $\alpha_{m,K}$ as both $m$ and $K$ grow could plausibly give an $\alpha$ with all $B(2^k\alpha)$ finite; if such a limit were shown to exist, it would be a B2LC counterexample, and the paper's own construction is the natural starting point.
  • The paper's obstruction at odd $m$ suggests the true extremal constant for B2LC, if a counterexample exists, might be an odd number; this is a heuristic from the $m=3$ and $m=5$ examples, not a claim of the paper.
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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

3 major / 5 minor

Summary. The paper studies quantitative lower bounds for the 2-adic Littlewood conjecture (2LC) and for a limsup variant (B2LC). For the sup-norm M, the authors prove sup_{k≥0} M(2^k α) ≥ 15 using a Sagemath exhaustive search, and derive the corresponding Diophantine infimum bound 1/15. For the limsup quantity B(α)=limsup a_n(α), they prove the main new result that sup_{k≥0} B(2^k α) ≥ 5 for every irrational α, using Hurwitz's continued-fraction multiplication-by-2 algorithm. They also construct, for every odd m≥3 and every K, quadratic irrationals α_{m,K} with B(2^k α_{m,K})=m for all 0≤k≤K, and they characterize all α with B(α)≤2 and B(2α)≤2.

Significance. If the main B-bound is correct, it is a new lower bound for the B-variant of 2LC and improves on the trivial C=2 bound, and it is proved by an appealing combination of Hurwitz's algorithm and equivalence arguments. The paper is honest about the relation to Badziahin's more recent M-bound 1/25, and the construction of arbitrarily long chains of equivalent shifts (Lemma 10, Theorem 4) is a useful contribution in its own right. However, two load-bearing parts need attention before the claims are fully established: the proof of Theorem 6 is presented too tersely and requires a nontrivial case analysis that is not written down, and the proof of Theorem 2 depends on a two-hour Sagemath run with no certificate or machine-readable verification. The writing is generally careful and the lemmas on Hurwitz's algorithm are a strength.

major comments (3)
  1. [Section 7, Theorem 6] The sentence 'For 4α not ∼ [3;1,1], this follows from Lemmas 14, 15 and 16' is not a valid inference as written. The lemmas only yield lower bounds on B(2^k α) for k in {0,1,3,4} from hypotheses on B(4α); they do not directly contradict sup_{k≥0}B(2^k α)≤4. A complete proof needs an explicit case analysis: first, if B(2^j α)=4 for some j≥2, applying Lemma 16 to 2^{j-2}α gives a shift with B≥5, so all B(2^j α) for j≥2 are at most 3; second, if B(2^j α)=3 for some j≥4, applying Lemma 15 to 2^{j-2}α gives a shift with B≥4 among indices ≥2, unless 2^j α∼[3;1,1], in which case B(2^{j+1}α)=8; hence B(2^j α)≤2 for j≥4; finally, Lemma 14 applied to 2^5 α contradicts B(2^4 α),B(2^6 α)≤2. This case analysis should be written out, including the special case 4α∼[3;1,1], which is currently dispatched with a one-line appeal to Lemma 9 and Examples 1 and 2.
  2. [Section 3, Theorem 2 and Remark 2] The proof of Theorem 2 rests entirely on a Sagemath exhaustive search for C=14, but no certificate, commit hash, machine-readable log, or independent verifier is provided. Since a single bug in the interval arithmetic, pruning, or the digit-min inequality would invalidate the theorem and Corollary 1, this is load-bearing. The authors should supply version-controlled code with a fixed commit hash, the exact input and output, and preferably a verifier that checks the exclusion certificates independently of the search that produced them.
  3. [Section 7, Lemma 15] In the proof of Lemma 15, after deriving that every sufficiently late digit 3 in 4α is followed by 1,1, the text says 'By extension, the same holds for the digits of 2α and 8α.' This is not justified by the preceding argument, which was carried out only for digits 3 of 4α, and the subsequent contradiction uses the rule for a digit 3 produced in 2α or 8α by Algorithm A. The step needs a proof, or the argument must be reorganized so that only the rule for 4α is used.
minor comments (5)
  1. [Abstract and Section 1] The abstract and the introduction state that the paper improves Badziahin's bound from 1/9 to 1/15, while the footnote and Section 3 acknowledge that Badziahin's later bound 1/25 is stronger; the abstract should be updated to reflect the current status.
  2. [Section 6, Lemma 9 proof] In the proof of Lemma 9, the citation 'by Lemma 11' appears to be a cross-reference error; the statement that α∼α/2 implies α is quadratic is Lemma 4, not Lemma 11.
  3. [Section 7, Remark 6] The notation '[5, 2, 1, 2]' and '[5; 1, 2, 2, 1]' is inconsistent; the first should presumably read '[5; 2, 1, 2]'.
  4. [Throughout] There are several typographical errors, including 'Bazdiahin' for 'Badziahin' in Section 2, 'availble' in Remark 2, 'RELA TED' in the title, and 'contin ued' in the references section; these should be corrected in the final version.
  5. [Section 3, Theorem 3] The bound 'n·9^k' in the proof of Theorem 3 is unnecessarily large; Lemma 8 would suffice with a factor of about 3^k, but the argument is otherwise fine.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's bounds rest on a reproducible exhaustive search and on direct lemmas, not on fitted parameters, self-citations, or definitions that presuppose the conclusions.

full rationale

The paper's main claims are Theorem 2 (sup_k M(2^k α) ≥ 15) and Theorem 6 (sup_k B(2^k α) ≥ 5). Theorem 2 is established by an explicit prefix-exclusion algorithm run in Sagemath for C = 14; the theorem is not defined in terms of the algorithm's output, and the algorithm is a finite computation with linked code rather than a fitted parameter or a self-referential construction. Theorem 6 rests on Lemmas 14–16, which are proved by direct analysis of Hurwitz's digit-sliding algorithm; no lemma assumes the target lower bound, and no external citation by the present authors is load-bearing. The paper explicitly acknowledges limitations that are correctness risks rather than circularity: Remark 2 gives only a GitHub link without a commit hash, machine-readable log, or certificate for the two-hour computation, and the inference in the proof of Theorem 6 that the case B(4α)<4 'follows from Lemmas 14, 15 and 16' is not fully justified as written. Remark 6 likewise states the authors could not push the bound beyond 5. None of these limitations makes a 'prediction' reduce to its input by construction. The B2LC and 2LC formulations are introduced as conjectures and are not used to define the constants 15 or 5. Self-citations are absent from the derivation chain; the cited works of Badziahin, Aka–Shapira, Hurwitz, and others are independent external results or classical algorithms. Accordingly, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities. The proof relies on standard continued fraction facts and on the correctness of a computer-assisted exhaustive search.

assumptions (4)
  • standard math Standard continued fraction facts: best approximation property, Legendre's theorem, and inequality (1).
    Used throughout Section 4 and in Corollary 1 as background; standard results.
  • standard math Hurwitz multiplication identities and the sliding-window lemmas (Lemmas 5-8).
    Proven in the paper, but they are the engine of the B-bound; they are algebraic identities plus parity bookkeeping.
  • standard math Endpoint digit-min monotonicity in the prefix-exclusion algorithm.
    Theorem 2 step (b) asserts that the next partial quotient over an interval is bounded below by the minimum of the endpoint next partial quotients; this follows from monotonicity of complete quotients but is not stated as a lemma.
  • ad hoc to paper Correctness of the Sagemath exhaustive search for C=14.
    Load-bearing for Theorem 2; code URL and timing are provided, but no formal certificate, commit hash, or execution logs are published.

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Cite this review

Pith. "Pith review of Some Bounds Related to the $2$-adic Littlewood Conjecture." pith.science (2026). https://pith.science/paper/KCLAXVO7

@misc{pith2026250604110,
  author       = {Pith},
  title        = {Pith review of: Some Bounds Related to the $2$-adic Littlewood Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCLAXVO7}},
  note         = {Machine review of arXiv:2506.04110}
}
abstract

For every irrational real $\alpha$, let $M(\alpha) = \sup_{n\geq 1} a_n(\alpha)$ denote the largest partial quotient in its continued fraction expansion (or $\infty$, if unbounded). The $2$-adic Littlewood conjecture (2LC) can be stated as follows: There exists no irrational $\alpha$ such that $M(2^k \alpha)$ is uniformly bounded by a constant $C$ for all $k\geq 0$. In 2016, Badziahin proved (considering a different formulation of 2LC) that if a counterexample exists, then the bound $C$ is at least $8$. We improve this bound to $15$. Then we focus on a ``B-variant'' of 2LC, where we replace $M(\alpha)$ by $B(\alpha) = \limsup_{n\to \infty} a_n(\alpha)$. In this setting, we prove that if $B(2^k \alpha) \leq C$ for all $k\geq 0$, then $C \geq 5$. For the proof we use Hurwitz's algorithm for multiplication of continued fractions by 2. Along the way, we find families of quadratic irrationals $\alpha$ with the property that for arbitrarily large $K$ there exist $\beta, 2\beta, 4 \beta, \ldots, 2^K \beta$ all equivalent to $\alpha$.

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Reference graph

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