An Improved Upper Bound for the Dirichlet Spectrum in Diophantine Approximation

Ethan Wang; Siyuan Wang; Zixuan Peng

arxiv: 2605.21275 · v1 · pith:2IUXD4VQnew · submitted 2026-05-20 · 🧮 math.NT

An Improved Upper Bound for the Dirichlet Spectrum in Diophantine Approximation

Zixuan Peng , Siyuan Wang , Ethan Wang This is my paper

Pith reviewed 2026-05-21 04:02 UTC · model grok-4.3

classification 🧮 math.NT

keywords Dirichlet spectrumDiophantine approximationCantor setsthicknesspartial quotientsray-origin constantsum-set theorems

0 comments

The pith

The ray-origin constant δ in the Dirichlet spectrum satisfies δ ≤ 111(397 + √26565)/65522 ≈ 0.94866

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

This paper improves the known upper bound on the ray-origin constant δ for the continuous part of the Dirichlet spectrum in Diophantine approximation. The authors construct a Cantor-type set F4* by restricting the possible partial quotients in continued fraction expansions. They prove that the thickness of the logarithm of this set exceeds 1, which combines with sum-set theorems to show that the product of the set with itself fills an entire interval. This interval property then produces the new numerical bound on δ. A reader would care because the result narrows the range of possible values that approximation constants can take for irrational numbers.

Core claim

We introduce the Cantor-type set F4* defined by certain restrictions on partial quotients. Proving that the thickness τ(log(F4*)) > 1 allows us to apply sum-set results and conclude that F4* · F4* is an interval. This establishes the new upper bound δ ≤ 111(397 + √26565)/65522 ≈ 0.94866 for the ray-origin constant in the Dirichlet spectrum.

What carries the argument

The Cantor-type set F4* defined by restrictions on partial quotients, whose log-thickness greater than 1 forces the product set F4* · F4* to be an interval via sum-set theorems

If this is right

The continuous part of the Dirichlet spectrum is contained in a strictly smaller interval than before
The refined Cantor set construction improves the bound obtained by earlier applications of Ivanov's method
The same thickness-plus-sum-set strategy can be reused with different partial-quotient restrictions to seek further improvements

Where Pith is reading between the lines

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

If the thickness condition can be verified for even more restrictive families of partial quotients, the resulting upper bound on δ could be lowered still further
The interval-product technique may transfer to other metric problems in Diophantine approximation that involve products or sums of Cantor sets
Numerical computation of the thickness for this explicit F4* could give an independent check on the analytic proof

Load-bearing premise

The central argument depends on proving that the thickness τ(log(F4*)) > 1 for the specific Cantor-type set F4*

What would settle it

Finding a concrete irrational number whose associated Diophantine constant lies strictly above 0.94866 but below the prior published bound would disprove the new upper bound

read the original abstract

We study the continuous part of the Dirichlet spectrum $\mathbb{D}$ and improve the best previously published upper bound for the ray-origin constant $\delta$. Building on and refining V. A. Ivanov's approach, we introduce a Cantor-type set $F_4^*$ defined by certain restrictions on partial quotients. For its thickness, we prove $\tau(\log(F_4^*))>1$, and apply sum-set results for Cantor sets to prove that the set $F_4^* \cdot F_4^*$ is an interval. Finally, we establish a new upper bound $\delta\le \frac{111(397+\sqrt{26565})}{65522}\approx0.94866$.

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 refines a Cantor set to push the upper bound on the ray-origin constant δ down to about 0.94866, with the thickness proof as the main new technical step.

read the letter

Colleague, This paper improves the upper bound on the ray-origin constant in the Dirichlet spectrum to δ ≤ 111(397 + √26565)/65522 ≈ 0.94866. That's the headline result. They introduce a Cantor-type set called F4* defined by restrictions on continued fraction partial quotients. The key step is proving that the thickness of the logarithm of this set is greater than 1. From there, sum-set theorems imply that the product F4* · F4* is an interval, which in turn gives the improved bound on the spectrum constant. What they do well is take an existing method and refine the set definition to get a measurable improvement in the bound. The explicit numerical value is new and could serve as a benchmark for future work in this area. The chain from construction to thickness to interval to bound is laid out clearly. The soft spot is the thickness estimate. It comes from calculating the infimum over all gaps of the ratio between the length of the adjacent component and the gap size. Because the set is built by forbidding certain patterns, there is a worst-case level where this ratio is smallest. If the calculation of that minimal ratio is off by even a little, it might not exceed 1, and the sum-set conclusion would not apply. This is the part that would need verification in a review. This kind of paper is for researchers who work on Diophantine approximation constants and the geometry of continued fractions. Someone already reading papers on the Dirichlet spectrum would get direct value from the new bound and the construction. It has a verifiable new result and uses established techniques without obvious circularity, so it should go to peer review rather than being rejected at the desk. Recommendation: yes, send it out for referees.

Referee Report

1 major / 1 minor

Summary. The paper improves the upper bound on the ray-origin constant δ in the Dirichlet spectrum by constructing a Cantor-type set F_4^* restricted by partial quotients, proving its log-thickness exceeds 1, applying sum-set theorems to establish that the product F_4^* · F_4^* is an interval, and deriving the bound δ ≤ 111(397+√26565)/65522 ≈ 0.94866.

Significance. This provides a better explicit upper bound for δ, building on Ivanov's approach. The use of thickness to guarantee an interval via sumsets is a standard technique in Diophantine approximation, and confirming the thickness >1 would make this a solid contribution to bounding the spectrum.

major comments (1)

[Section proving thickness of log(F_4^*)] The central claim rests on τ(log(F_4^*)) > 1. The manuscript estimates gap ratios in the Cantor construction; the infimum must be shown >1. Please provide the explicit value of the minimal ratio and confirm the length calculations for the intervals and gaps at the level where the smallest ratio occurs, as this directly determines whether the sum-set theorem applies to conclude that the product set is an interval.

minor comments (1)

[Abstract] It would be helpful to state the previous best known upper bound for comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment point by point below and are prepared to revise the paper to incorporate the requested clarifications.

read point-by-point responses

Referee: [Section proving thickness of log(F_4^*)] The central claim rests on τ(log(F_4^*)) > 1. The manuscript estimates gap ratios in the Cantor construction; the infimum must be shown >1. Please provide the explicit value of the minimal ratio and confirm the length calculations for the intervals and gaps at the level where the smallest ratio occurs, as this directly determines whether the sum-set theorem applies to conclude that the product set is an interval.

Authors: We appreciate this comment, which correctly identifies that the thickness proof relies on verifying the infimum of the gap ratios exceeds 1. In the current manuscript, the gap ratios are estimated iteratively for the restricted partial quotients defining F_4^*, with the construction ensuring that removed intervals are sufficiently small relative to the remaining components. However, we agree that an explicit statement of the minimal ratio and the precise interval/gap lengths at the critical stage would improve clarity and directly support the application of the sum-set theorem. In the revised version, we will add a dedicated subsection or appendix computing these quantities stage by stage, confirming the infimum is strictly greater than 1 (achieved at an early finite level of the construction) and verifying the length relations. This does not alter the main result but makes the argument fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses explicit Cantor-set construction and external sum-set theorems

full rationale

The paper defines F_4^* via explicit restrictions on continued-fraction partial quotients, proves τ(log(F_4^*)) > 1 by direct gap-ratio estimates at each construction level, invokes cited sum-set theorems (independent of the present work) to obtain that F_4^* · F_4^* is an interval, and deduces the numerical upper bound on δ from that interval property. None of these steps reduces by definition or by self-citation to the target bound; the thickness lower bound is obtained from finite forbidden-pattern analysis rather than from fitting or renaming. The cited sum-set results are treated as external benchmarks, satisfying the self-contained criterion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard mathematical background in continued fractions, Cantor sets, and thickness theory. No free parameters are fitted to data. The new set F4* is a constructed object rather than an invented physical entity.

axioms (1)

standard math Standard properties of continued fraction expansions and metric properties of Cantor sets in Diophantine approximation hold.
Invoked throughout the construction of F4* and the thickness argument.

pith-pipeline@v0.9.0 · 5644 in / 1273 out tokens · 35557 ms · 2026-05-21T04:02:03.462130+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 embed_injective, embed_strictMono_of_one_lt echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We introduce a Cantor-type set F4* defined by certain restrictions on partial quotients. For its thickness, we prove τ(log(F4*))>1, and apply sum-set results for Cantor sets to prove that the set F4*·F4* is an interval.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

δ = inf{d : (d,1] ⊂ D} and μ = inf{z : [z,∞) ⊂ Ω} with δ = μ/(1+μ)

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

8 extracted references · 8 canonical work pages

[1]

Akhunzhanov and D

R. Akhunzhanov and D. Shatskov,On Dirichlet spectrum for two-dimensional simultaneous Diophantine approximation, Mosc. J. Comb. Number Theory 3, No. 3-4, 5-23 (2013)

work page 2013
[2]

Diviˇ s and B

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work page 1971
[3]

Hall, Jr.,On the sum and product of continued fractions, Ann

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[4]

V. A. Ivanov,Origin of the ray in the Dirichlet spectrum of a problem in the theory of diophantine approximations, J Math. Sci. 19, 1169–1183 (1982)

work page 1982
[5]

Lesca,Sur les approximations diophantiennes a une dimension, L’Universite de Grenoble, PhD Thesis (1968)

J. Lesca,Sur les approximations diophantiennes a une dimension, L’Universite de Grenoble, PhD Thesis (1968)

work page 1968
[6]

Morimoto,Zur Theorie der Approximation einer irrationalen Zahl durch rationale Zahlen, Tohoku Math

S. Morimoto,Zur Theorie der Approximation einer irrationalen Zahl durch rationale Zahlen, Tohoku Math. J., 45 (1938), 177—187

work page 1938
[7]

Pitcyn,On the discrete part of the Dirichlet spectrum, Ramanujan J

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work page 2025
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Szekeres,On a problem of the lattice plane, J

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[1] [1]

Akhunzhanov and D

R. Akhunzhanov and D. Shatskov,On Dirichlet spectrum for two-dimensional simultaneous Diophantine approximation, Mosc. J. Comb. Number Theory 3, No. 3-4, 5-23 (2013)

work page 2013

[2] [2]

Diviˇ s and B

B. Diviˇ s and B. Nov´ ak,A remark on the theory of diophantine approximations, Comment. Math. Univ. Carolinae, 12, No.1, 127-141 (1971)

work page 1971

[3] [3]

Hall, Jr.,On the sum and product of continued fractions, Ann

M. Hall, Jr.,On the sum and product of continued fractions, Ann. of Math. (2), 48, 1947, pp. 966–993

work page 1947

[4] [4]

V. A. Ivanov,Origin of the ray in the Dirichlet spectrum of a problem in the theory of diophantine approximations, J Math. Sci. 19, 1169–1183 (1982)

work page 1982

[5] [5]

Lesca,Sur les approximations diophantiennes a une dimension, L’Universite de Grenoble, PhD Thesis (1968)

J. Lesca,Sur les approximations diophantiennes a une dimension, L’Universite de Grenoble, PhD Thesis (1968)

work page 1968

[6] [6]

Morimoto,Zur Theorie der Approximation einer irrationalen Zahl durch rationale Zahlen, Tohoku Math

S. Morimoto,Zur Theorie der Approximation einer irrationalen Zahl durch rationale Zahlen, Tohoku Math. J., 45 (1938), 177—187

work page 1938

[7] [7]

Pitcyn,On the discrete part of the Dirichlet spectrum, Ramanujan J

S. Pitcyn,On the discrete part of the Dirichlet spectrum, Ramanujan J. 67, No. 4, Paper No. 104, 17 p. (2025)

work page 2025

[8] [8]

Szekeres,On a problem of the lattice plane, J

G. Szekeres,On a problem of the lattice plane, J. London Math. Soc. 12 (1937), 88 – 93. 15

work page 1937