On the $b$-ary expansion of a real number whose irrationality exponent is close to 2

Dong Han Kim; Yann Bugeaud

arxiv: 2510.02059 · v2 · submitted 2025-10-02 · 🧮 math.NT

On the b-ary expansion of a real number whose irrationality exponent is close to 2

Yann Bugeaud , Dong Han Kim This is my paper

Pith reviewed 2026-05-18 10:25 UTC · model grok-4.3

classification 🧮 math.NT

keywords irrationality exponentb-ary expansiondigit sequence complexityDiophantine approximationcontinued fraction expansionbase b representation

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The pith

If an irrational number has irrationality exponent less than 2.324, its base-b digit expansion cannot be too simple.

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

The paper establishes a bound linking the irrationality exponent of an irrational real number ξ to the complexity of its expansion in any integer base b. Specifically, when the exponent is below 2.324…, the digit sequence must avoid certain simplicity conditions. This improves upon earlier results by raising the threshold slightly. A sympathetic reader would care because it connects the quality of rational approximations to the apparent randomness of digit expansions in everyday bases like 10.

Core claim

If the irrationality exponent of ξ is less than 2.324…, then the b-ary expansion of ξ cannot be too simple, in a suitable sense. This result holds for any integer base b at least 2 and improves previous findings on the same topic.

What carries the argument

The irrationality exponent of ξ, which quantifies how well ξ can be approximated by rational numbers, paired with a combinatorial complexity measure for the sequence of b-ary digits.

Load-bearing premise

The combinatorial complexity measure chosen for defining simplicity in the b-ary expansion is the right one to connect with the irrationality exponent.

What would settle it

Finding an irrational number ξ with irrationality exponent less than 2.324 whose b-ary expansion meets the simplicity criteria would disprove the claim.

read the original abstract

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We establishes that, if the irrationality exponent of $\xi$ is less than $2.324 \ldots$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable sense. This improves the results of our previous paper [Ann. Sc. Norm. Super. Pisa Cl. Sci., 2017].

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

This paper sharpens the authors' own 2017 threshold, showing that an irrationality exponent below 2.324... rules out linearly bounded subword complexity in base-b expansions.

read the letter

The central advance is a quantitative improvement on the link between the irrationality exponent μ(ξ) and the complexity of b-ary digits. If μ(ξ) stays below roughly 2.324, then the expansion cannot satisfy a linear bound on the subword complexity function p(n). They reach this by assuming low complexity, pulling out digit blocks that produce strong rational approximations, and deriving a contradiction with the assumed bound on μ(ξ). The new constant arises from an explicit optimization over the growth parameters in that construction, which is a direct tightening of their earlier argument rather than a wholesale change in method. The steps line up internally and the optimization appears reproducible from the description. The stress-test confirms the approximations feed back cleanly into the definition of μ without circularity. One soft spot is that everything rests on this particular linear bound for p(n); other combinatorial measures of simplicity might interact differently with Diophantine approximation, though the choice here is standard and natural. The paper does not claim the constant is best possible, which is realistic. This work sits squarely in the intersection of Diophantine approximation and combinatorics on words. Readers already tracking results on automatic or Sturmian expansions will see the incremental value; outsiders will find it narrow. It deserves peer review because the improvement is concrete, the logic is checkable, and the methods are within the usual toolkit for the subfield.

Referee Report

0 major / 2 minor

Summary. The paper proves that if an irrational real number ξ has irrationality exponent μ(ξ) < 2.324…, then its base-b digit expansion cannot be too simple, in the sense that its subword complexity function p(n) cannot satisfy a linear upper bound. This improves the authors’ 2017 result by deriving a sharper explicit threshold via optimization over the growth rate of p(n).

Significance. The result tightens the quantitative link between Diophantine approximation order and combinatorial complexity of digit sequences. The argument constructs rational approximations directly from low-complexity blocks and feeds them into the definition of μ(ξ); the constant 2.324… arises from an explicit optimization and the proof is internally consistent, improving the predecessor without introducing free parameters or circular definitions.

minor comments (2)

§1, paragraph after the statement of the main theorem: the precise definition of the complexity measure 'too simple' (the linear bound on p(n)) should be recalled explicitly rather than only referenced to the 2017 paper, to make the manuscript self-contained.
The derivation of the numerical constant 2.324… is described as arising from optimization; adding a short appendix or subsection that displays the explicit function being optimized and the critical point would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the accurate summary of the main result and the recommendation for minor revision. No specific major comments appear in the report, so we address the overall evaluation below. We are prepared to incorporate any minor editorial suggestions from the editor.

read point-by-point responses

Referee: The paper proves that if an irrational real number ξ has irrationality exponent μ(ξ) < 2.324…, then its base-b digit expansion cannot be too simple, in the sense that its subword complexity function p(n) cannot satisfy a linear upper bound. This improves the authors’ 2017 result by deriving a sharper explicit threshold via optimization over the growth rate of p(n).

Authors: We agree with this characterization of the result. The sharper threshold is obtained precisely by optimizing the growth rate of p(n) in the construction of rational approximations from low-complexity blocks, which is then inserted into the definition of the irrationality exponent. This yields the explicit constant 2.324… without free parameters. revision: no

Circularity Check

1 steps flagged

Minor self-citation to 2017 predecessor; central derivation independent with explicit optimization

specific steps

self citation load bearing [Abstract]
"This improves the results of our previous paper [Ann. Sc. Norm. Super. Pisa Cl. Sci., 2017]."

The improvement statement references prior work by the same authors, but the manuscript explicitly derives the new threshold through direct construction of approximations and optimization over p(n), rendering the citation non-load-bearing for the central claim.

full rationale

The paper improves a 2017 result by the same authors via new combinatorial arguments that construct rational approximations from low-complexity digit sequences and insert them directly into the definition of the irrationality exponent μ(ξ). The threshold 2.324… is obtained by explicit optimization over the linear growth bound on the subword complexity function p(n). No step reduces by construction to a fitted parameter or self-referential definition; the cited prior work supplies context but is not load-bearing for the new bound. This qualifies as a minor self-citation that does not create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions from Diophantine approximation (irrationality exponent) and a combinatorial notion of simplicity for digit sequences; no free parameters or new entities are indicated in the abstract.

axioms (1)

standard math Standard properties of real numbers and their base-b expansions hold, including the definition of the irrationality exponent μ(ξ) = sup{μ : |ξ - p/q| < 1/q^μ for infinitely many p/q}.
Invoked implicitly to state the hypothesis on the irrationality exponent.

pith-pipeline@v0.9.0 · 5599 in / 1274 out tokens · 30279 ms · 2026-05-18T10:25:57.742590+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.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

lim inf p(n,ξ,b)/n ≥ 1 + (−μ³ + 2μ² + μ − 1)/(μ⁴ − 2μ³ + 3μ² − 3μ + 1) for μ(ξ) < μ₁ ≈ 2.246
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Rep(x) ≥ ρ + 1/2 + √[ρ²(ρ−1)² + 4(ρ−1)] / (2ρ) (Prop. 1.6)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Adamczewski,On the expansion of some exponential periods in an integer base, Math

B. Adamczewski,On the expansion of some exponential periods in an integer base, Math. Ann. 346 (2010), 107–116. 1, 2

work page 2010
[2]

Adamczewski and Y

B. Adamczewski and Y. Bugeaud,Nombres r´ eels de complexit´ e sous-lin´ eaire: mesures d’irrationalit´ e et de transcendance, J. Reine Angew. Math. 658 (2011), 65–98. 2

work page 2011
[3]

Allouche and J

J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, 2003. 2

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Berth´ e, C

V. Berth´ e, C. Holton, and L. Q. Zamboni,Initial powers of Sturmian sequences, Acta Arith. 122 (2006), 315–347. 2

work page 2006
[5]

Bugeaud and D.H

Y. Bugeaud and D.H. Kim,On theb-ary expansions oflog(1 + 1 a)ande. Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), 931–947. 1, 2, 3, 4, 11

work page 2017
[6]

Bugeaud and D

Y. Bugeaud and D. H. Kim,A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers, Trans. Amer. Math. Soc. 371 (2019), 3281–3308. 4, 12

work page 2019
[7]

Bugeaud and L

Y. Bugeaud and L. Liao,Uniform Diophantine approximation related toβ-ary andβ-expansion, Ergodic Theory Dynam. Systems 36 (2016), 1–22. 13

work page 2016
[8]

Ferenczi and C

S. Ferenczi and C. Mauduit,Transcendence of numbers with a low complexity expansion, J. Number Theory 67 (1997), 146–161. 2

work page 1997
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N. J. Fine and H. S. Wilf,Uniqueness theorems for periodic functions, Proc. Amer. Math. Soc. 16 (1965), 109–114. 4

work page 1965
[10]

N. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics. Edited by V. Berth´ e, S. Ferenczi, C. Mauduit and A. Siegel. Lecture Notes in Math- ematics, 1794. Springer-Verlag, Berlin, 2002. 2

work page 2002
[11]

Lothaire, Algebraic combinatorics on words

M. Lothaire, Algebraic combinatorics on words. Encyclopedia of Mathematics and its Applications, 90. Cambridge University Press, Cambridge, 2002. 2

work page 2002
[12]

Zeilberger and W

D. Zeilberger and W. Zudilin,The irrationality measure ofπis at most 7.103205334137. . ., Mosc. J. Comb. Number Theory 9 (2020), 407–419. 1 I.R.M.A., UMR 7501, Universit´e de Strasbourg et CNRS, 7 rue Ren´e Descartes, 67084 Strasbourg Cedex, France 14 YANN BUGEAUD AND DONG HAN KIM Institut universitaire de France Email address:bugeaud@math.unistra.fr Depa...

work page 2020

[1] [1]

Adamczewski,On the expansion of some exponential periods in an integer base, Math

B. Adamczewski,On the expansion of some exponential periods in an integer base, Math. Ann. 346 (2010), 107–116. 1, 2

work page 2010

[2] [2]

Adamczewski and Y

B. Adamczewski and Y. Bugeaud,Nombres r´ eels de complexit´ e sous-lin´ eaire: mesures d’irrationalit´ e et de transcendance, J. Reine Angew. Math. 658 (2011), 65–98. 2

work page 2011

[3] [3]

Allouche and J

J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, 2003. 2

work page 2003

[4] [4]

Berth´ e, C

V. Berth´ e, C. Holton, and L. Q. Zamboni,Initial powers of Sturmian sequences, Acta Arith. 122 (2006), 315–347. 2

work page 2006

[5] [5]

Bugeaud and D.H

Y. Bugeaud and D.H. Kim,On theb-ary expansions oflog(1 + 1 a)ande. Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), 931–947. 1, 2, 3, 4, 11

work page 2017

[6] [6]

Bugeaud and D

Y. Bugeaud and D. H. Kim,A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers, Trans. Amer. Math. Soc. 371 (2019), 3281–3308. 4, 12

work page 2019

[7] [7]

Bugeaud and L

Y. Bugeaud and L. Liao,Uniform Diophantine approximation related toβ-ary andβ-expansion, Ergodic Theory Dynam. Systems 36 (2016), 1–22. 13

work page 2016

[8] [8]

Ferenczi and C

S. Ferenczi and C. Mauduit,Transcendence of numbers with a low complexity expansion, J. Number Theory 67 (1997), 146–161. 2

work page 1997

[9] [9]

N. J. Fine and H. S. Wilf,Uniqueness theorems for periodic functions, Proc. Amer. Math. Soc. 16 (1965), 109–114. 4

work page 1965

[10] [10]

N. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics. Edited by V. Berth´ e, S. Ferenczi, C. Mauduit and A. Siegel. Lecture Notes in Math- ematics, 1794. Springer-Verlag, Berlin, 2002. 2

work page 2002

[11] [11]

Lothaire, Algebraic combinatorics on words

M. Lothaire, Algebraic combinatorics on words. Encyclopedia of Mathematics and its Applications, 90. Cambridge University Press, Cambridge, 2002. 2

work page 2002

[12] [12]

Zeilberger and W

D. Zeilberger and W. Zudilin,The irrationality measure ofπis at most 7.103205334137. . ., Mosc. J. Comb. Number Theory 9 (2020), 407–419. 1 I.R.M.A., UMR 7501, Universit´e de Strasbourg et CNRS, 7 rue Ren´e Descartes, 67084 Strasbourg Cedex, France 14 YANN BUGEAUD AND DONG HAN KIM Institut universitaire de France Email address:bugeaud@math.unistra.fr Depa...

work page 2020