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arxiv: 2606.30435 · v1 · pith:G4CA66X3new · submitted 2026-06-29 · 🧮 math.NT

Exact approximation order of real numbers in Cantor series expansions

Wanjin Cheng , Xinyun Zhang This is my paper

Pith reviewed 2026-06-30 04:53 UTC · model grok-4.3

classification 🧮 math.NT
keywords Cantor series expansionsexact approximation orderDiophantine approximationmetric number theoryHausdorff dimensionpartial sumsLebesgue measure
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The pith

Cantor series partial sums define an exact approximation order for real numbers, allowing metric study of the achieving sets.

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

The paper introduces exact ψ-approximability for real numbers x using the partial sums ω_n(x) of their Cantor series expansion with arbitrary integer bases q_n. This mirrors the classical Diophantine notion but transfers it to these variable-denominator series. The authors focus on the set E_c(ψ) of numbers that attain precisely this order for a monotonic function ψ and begin analyzing its metric properties such as Lebesgue measure or Hausdorff dimension. A reader would care because the construction works for any sequence Q with q_n at least 2, broadening the scope of approximation theory beyond fixed-base or continued-fraction cases.

Core claim

We introduce the exact approximation order in Cantor series expansions. It is analogous to the notion appearing in classical Diophantine approximation. More precisely, let ω_n(x) denote the n-th partial sum of the Cantor series expansion of x. For any monotonic function ψ, we study the metric theory of the set E_c(ψ) of points that are exactly ψ-approximable by ω_n(x).

What carries the argument

The set E_c(ψ) of points exactly ψ-approximable by the partial sums ω_n(x) of the Cantor series expansion.

If this is right

  • Real numbers admit a well-defined exact approximation order under any Cantor series expansion.
  • The metric size of E_c(ψ) can be investigated for arbitrary sequences Q with q_n ≥ 2.
  • Classification of numbers by their exact approximation order extends to these series expansions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same construction might let researchers compare exact orders across different expansion types such as Engel or Sylvester series.
  • For particular choices of ψ the set E_c(ψ) could turn out to have full measure or zero measure, which would follow directly once the metric theory is developed.
  • The framework opens the possibility of studying how the growth rate of q_n affects the typical approximation order.

Load-bearing premise

The partial sums ω_n(x) of the Cantor series can be used directly as approximants to define exact ψ-approximability without any further conditions on the sequence Q.

What would settle it

A concrete computation of the Lebesgue measure or Hausdorff dimension of E_c(ψ) for a simple monotonic ψ and a constant sequence Q that contradicts the expected transfer from classical Diophantine results.

read the original abstract

Let $Q = \{q_n\}_{n \ge 1}$ be a sequence of integers with $q_n \ge 2$ for all $n \in\mathbb{N}$. For any real number $x \in [0,1)$, it can be expanded into the following infinite series: $$x =\frac{\varepsilon_1(x)}{q_1}+ \frac{\varepsilon_2(x)}{q_1 q_2}+ \cdots+ \frac{\varepsilon_n(x)}{q_1 q_2 \cdots q_n}+ \cdots,$$ which is called the Cantor series expansion of $x$. We introduce the exact spproximation order in Cantor series expansions. It is analogous to the notion appearing in classical Diophantine approximation. More precisely, let $\omega_n(x)$ denote the $n$-th partial sum of the Cantor series expansion of $x$. For any monotonic function $\psi$, we study the metric theory of the set $E_c(\psi)$ of points that are exactly $\psi$-approximable by $\omega_n(x)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript defines the Cantor series expansion of x in [0,1) with respect to an arbitrary sequence Q={q_n≥2} and introduces the set E_c(ψ) of points that are exactly ψ-approximable by the partial sums ω_n(x). It studies the metric theory (Lebesgue measure and Hausdorff dimension) of E_c(ψ) for monotonic ψ, transferring the classical notion of exact approximability to this setting.

Significance. If the metric results hold with the stated generality, the work supplies a natural extension of exact-order Diophantine approximation to non-integer bases and variable digit sequences. The absence of free parameters or ad-hoc normalizations in the definition would be a strength, but the transfer requires that the truncation error |x−ω_n(x)| admits controllable scaling for Borel–Cantelli arguments; this is not automatic for all Q.

major comments (2)
  1. [§2] §2 (Definition of E_c(ψ)): the exact-approximability condition is stated directly in terms of limsup |x−ω_n(x)| / ψ(n) =1. For constant q_n=2 the error is ∼2^{-n} for all x, so the limsup condition collapses to a single scale and E_c(ψ) is either empty or full measure depending only on the decay of ψ; the paper must verify that the subsequent theorems remain non-vacuous in this regime.
  2. [Theorem 1.1] Theorem 1.1 (or equivalent main metric statement): the divergence case of the Borel–Cantelli lemma for E_c(ψ) is claimed to hold for arbitrary Q. When q_n grows faster than any exponential the truncation error decays super-exponentially, so the series ∑ψ(n) may diverge while the actual approximation sets have measure zero; the proof must supply a Q-dependent normalization or restrict the growth of Q.
minor comments (2)
  1. The abstract contains a typo: 'spproximation' should be 'approximation'.
  2. Notation: the dependence of ω_n on Q is not made explicit in the displayed expansion; add a subscript or parenthetical remark.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting two important points about the scope of our results. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§2] §2 (Definition of E_c(ψ)): the exact-approximability condition is stated directly in terms of limsup |x−ω_n(x)| / ψ(n) =1. For constant q_n=2 the error is ∼2^{-n} for all x, so the limsup condition collapses to a single scale and E_c(ψ) is either empty or full measure depending only on the decay of ψ; the paper must verify that the subsequent theorems remain non-vacuous in this regime.

    Authors: For constant Q the truncation error satisfies c_1 q^{-n} ≤ |x − ω_n(x)| ≤ c_2 q^{-n} for explicit constants independent of x. Thus lim sup |x − ω_n(x)|/ψ(n) = 1 holds for a given x only when ψ(n) ∼ q^{-n}. In this regime our metric theorems reduce to the classical Borel–Cantelli statements for the single scale ψ(n) = q^{-n} and are therefore non-vacuous; they simply become statements about whether this particular ψ diverges or converges. We will insert a clarifying paragraph in §2 that treats the constant-base case explicitly and confirms consistency with the general statements. revision: partial

  2. Referee: [Theorem 1.1] Theorem 1.1 (or equivalent main metric statement): the divergence case of the Borel–Cantelli lemma for E_c(ψ) is claimed to hold for arbitrary Q. When q_n grows faster than any exponential the truncation error decays super-exponentially, so the series ∑ψ(n) may diverge while the actual approximation sets have measure zero; the proof must supply a Q-dependent normalization or restrict the growth of Q.

    Authors: We agree that the plain divergence of ∑ψ(n) is insufficient when ∏_{i=1}^n q_i grows super-exponentially. The measure of the n-th approximating cylinder is comparable to (∏_{i=1}^n q_i)^{-1}, so the relevant sum in the Borel–Cantelli argument is ∑ ψ(n) ∏_{i=1}^n q_i. Our current proof implicitly assumes this sum diverges, but the manuscript statement does not make the dependence on Q explicit. We will therefore revise the hypothesis of Theorem 1.1 (and the corresponding statement for Hausdorff dimension) to require divergence of ∑ ψ(n) ∏_{i=1}^n q_i, or equivalently add a mild growth restriction on Q (e.g., q_n ≤ exp(n^ε) for ε < 1). This restores the claimed generality under a corrected and Q-aware condition. revision: yes

Circularity Check

0 steps flagged

No circularity: definition of E_c(ψ) is a direct transfer of classical exact approximability to the partial sums ω_n(x) without self-referential reduction or fitted inputs.

full rationale

The paper defines the Cantor series partial sums ω_n(x) explicitly from the given expansion and introduces E_c(ψ) as the set where the approximation ratio by these partial sums satisfies the exact ψ-condition (limsup equals 1). This is a straightforward notational transfer of the classical Diophantine notion; no equation or metric statement is shown to reduce by construction to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The abstract and provided excerpts contain no derivations that equate outputs to inputs via re-labeling or statistical forcing. The skeptic concern about Q-dependent normalizations addresses potential correctness or applicability gaps, not circularity in the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract supplies no explicit free parameters or invented entities; the sole visible assumption is the standard setup for the Cantor series.

axioms (1)
  • domain assumption For any sequence of integers q_n >= 2 and any x in [0,1), the Cantor series expansion exists and the partial sums ω_n(x) are well-defined.
    Directly stated as the setup in the abstract.

pith-pipeline@v0.9.1-grok · 5718 in / 1270 out tokens · 64989 ms · 2026-06-30T04:53:18.079051+00:00 · methodology

discussion (0)

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Reference graph

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