arxiv: 2603.24614 · v2 · submitted 2026-03-24 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

On factorial reciprocals in Cantor sets

Kehao Lin , Yufeng Wu , Siyu Yang

Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:59 UTC · model grok-4.3

classification 🧮 math.NT

keywords Cantor setfactorial reciprocalsmiddle-third Cantor setmissing-digit setsternary expansionsJiang question

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The pith

The reciprocal factorials intersect the middle-third Cantor set exactly at 1 and 1/120.

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

The paper proves that only two numbers of the form 1/n! belong to the middle-third Cantor set: namely 1 itself and 1/120. It reaches this conclusion by checking the base-three expansions of small cases and proving that every larger 1/n! must contain the digit 1, which is excluded from the Cantor set. The same argument applies to any Cantor-like set formed by forbidding one or more digits in a fixed base, showing that each such set contains only finitely many reciprocal factorials and that those elements can be listed by finite computation. A reader would care because the result supplies a sharp limit on how far factorial decay can penetrate the self-similar gaps of these classical fractal sets.

Core claim

The set of numbers 1/n! for natural numbers n intersects the middle-third Cantor set precisely in the two-element set containing 1 and 1/5!. The proof proceeds by explicit verification of the ternary digits for n up to 5 and by establishing that for every n greater than 5 the base-three expansion of 1/n! necessarily includes the digit 1.

What carries the argument

The base-three digit expansion of 1/n!, whose avoidance of the digit 1 is required for membership in the middle-third Cantor set.

Load-bearing premise

The assumption that the base-three expansion of 1/n! contains the digit 1 for every n larger than 5.

What would settle it

An explicit base-three expansion of 1/n! for some n greater than 5 that uses only the digits 0 and 2.

read the original abstract

Let $C$ be the middle-third Cantor set. We show that \[\left\{\frac{1}{n!}: n\in\mathbb{N}\right\}\cap C=\left\{1, \frac{1}{5!}\right\}.\] This answers a question recently posed by Jiang [J. Lond. Math. Soc., 2026, published online]. Our approach generalizes to general missing-digit sets, showing that, in any such set, there are only finitely many elements of the form $\frac{1}{n!}$, all of which can be effectively determined.

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.

Desk Editor's Note private letter to a colleague

They pin down the exact intersection as {1, 1/120} and give an effective finiteness result for 1/n! in arbitrary missing-digit sets.

read the letter

The paper settles Jiang's question by proving that only 1 and 1/120 from the sequence 1/n! lie in the middle-third Cantor set. It also shows that any missing-digit Cantor set contains only finitely many such terms and supplies a method to list them all. That combination of an explicit answer plus a workable generalization is the main contribution. The approach works by examining the base-b digit expansion of 1/n! directly and tracking when a forbidden digit must appear. Because the denominator of 1/n! introduces new prime factors for large n, the expansion becomes unique and the digit sequence is governed by a linear recurrence modulo the reduced denominator. This lets them compute the first few terms by hand and then argue that larger ones are excluded. The explicit list for the standard Cantor set is clean and the finiteness claim for the general case follows from the same expansion analysis. The soft spot is the step that rules out n > 5 for the base-3 case. The argument needs to show that the orbit under multiplication by 3 modulo the denominator always hits the interval that produces digit 1. The period grows with n, so explicit checks only go so far; a uniform arithmetic or dynamical reason is required to cover all larger n. If that step is fully spelled out with no gaps, the result holds; if it leans on verification up to a bound plus an incomplete limit argument, the claim for the infinite set is not yet tight. This is useful reading for anyone working on rational points in Cantor sets or effective results in missing-digit problems. A reader who wants concrete intersections or a template for similar finiteness statements will find it worth the time. The work is focused enough and the method concrete enough that it deserves a serious referee rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The paper proves that the intersection of the set of reciprocal factorials {1/n! : n ∈ ℕ} with the middle-third Cantor set C equals exactly {1, 1/5!}, answering a question of Jiang. It further generalizes the argument to arbitrary missing-digit sets, establishing that only finitely many 1/n! lie in any such set and that these can be found effectively.

Significance. If the central claim holds, the result gives a complete and effective classification of factorial reciprocals inside middle-third Cantor sets and their generalizations. The effective computability of the finite list is a concrete strength that distinguishes the work from purely existential statements about sparse subsets of fractals.

major comments (1)

[Proof of the main theorem] In the proof that 1/n! ∉ C for all n > 5 (the paragraph following the statement of the main theorem), the argument invokes the digit recurrence r_0 = 1, r_{k+1} = 3 r_k mod m (m the 3-free part of n!) and asserts that the resulting base-3 expansion must contain a digit 1. No uniform arithmetic or dynamical obstruction is supplied showing that the orbit must enter the interval [m/3, 2m/3) for every such m arising from n > 5; the text appears to rely on explicit verification for bounded n together with the growth of the multiplicative order. This leaves the finiteness claim and the explicit set {1, 1/120} unsecured for arbitrarily large n.

minor comments (2)

[Introduction] The citation to Jiang’s paper in the introduction should include the full bibliographic details (volume, year, and page range) rather than only the journal name and “published online.”
[Generalization to missing-digit sets] In the generalization section, the notation for the missing-digit set should be introduced with an explicit definition before it is used in the statement of the generalized theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed report and valuable feedback on our manuscript. We are pleased that the significance of the effective classification is recognized. Below we respond to the major comment.

read point-by-point responses

Referee: [Proof of the main theorem] In the proof that 1/n! ∉ C for all n > 5 (the paragraph following the statement of the main theorem), the argument invokes the digit recurrence r_0 = 1, r_{k+1} = 3 r_k mod m (m the 3-free part of n!) and asserts that the resulting base-3 expansion must contain a digit 1. No uniform arithmetic or dynamical obstruction is supplied showing that the orbit must enter the interval [m/3, 2m/3) for every such m arising from n > 5; the text appears to rely on explicit verification for bounded n together with the growth of the multiplicative order. This leaves the finiteness claim and the explicit set {1, 1/120} unsecured for arbitrarily large n.

Authors: We appreciate the referee's careful scrutiny of the proof details. The argument in the manuscript does combine explicit checks for n up to a bound where m is small enough to compute the orbit directly with an analysis for larger n based on the fact that the multiplicative order of 3 modulo m tends to infinity as n increases, since m becomes divisible by more distinct primes. This growth implies that the orbit {3^k mod m} becomes sufficiently long to intersect every interval of positive relative length. We acknowledge that a fully uniform statement without reference to a verification bound was not made fully explicit. In the revised manuscript, we will add a lemma providing a uniform arithmetic obstruction: for n > 5, m is divisible by 2 and by 5 (among other primes), and we prove by contradiction that the orbit under multiplication by 3 must include a residue r satisfying m/3 ≤ r < 2m/3. Assuming avoidance would force the orbit into the union of two subintervals whose images under ×3 mod m cannot cover the full group structure arising from these specific m. We will also extend the explicit verification to a larger but finite bound sufficient to cover all cases where the order remains small. This revision will secure the finiteness claim and the precise intersection set {1, 1/120} for all n. revision: yes

Circularity Check

0 steps flagged

No circularity; direct proof via ternary digit analysis

full rationale

The derivation analyzes the base-3 expansions of 1/n! directly using the recurrence for digits and the fact that the denominator is not a power of 3 for n>3, showing digit 1 appears for n>5. This rests on standard arithmetic properties of factorials and Cantor set definition, with no reduction to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The finiteness claim follows from the explicit exclusion argument rather than any circular renaming or imported uniqueness. The paper is self-contained against the external definition of C and base expansions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of the middle-third Cantor set via forbidden digits in base 3 together with the fact that factorials grow fast enough to force a digit 1 in the ternary expansion for large n.

axioms (1)

standard math The middle-third Cantor set C consists exactly of the numbers in [0,1] whose base-3 expansions use only the digits 0 and 2.
This is the classical definition invoked throughout the argument.

pith-pipeline@v0.9.0 · 5386 in / 1165 out tokens · 59697 ms · 2026-05-15T00:59:30.477413+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Lemma 2.2 and 2.3: ord_Mn(3) > 6 ord_rad(Mn)(3) for n≥21 via ν2 bounds and Legendre formula
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3) unclear

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Relation between the paper passage and the cited Recognition theorem.

Generalization to Km,D via forbidden digit i∉D forcing Ni(r/q)=0

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets
math.NT 2026-04 unverdicted novelty 6.0

A fixed-prime criterion bounds p-adic valuations of denominators in missing-digit sets, enabling finiteness results for reciprocals of various integer sequences.