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The Dirichlet spectrum with respect to $L_1$ norm is $\left[\frac12,1\right]$

T0 review · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The Dirichlet spectrum with respect to L1 norm is the interval from 1/2 to 1.

desk verdict This paper settles the L1 Dirichlet spectrum as exactly [1/2,1] through explicit continued-fraction constructions for the Minkowski spectrum. read the letter →

arxiv 2606.08865 v1 pith:IRAUOVRU submitted 2026-06-07 math.NT

Nikita Shulga This is my paper · ORCID
classification math.NT
keywords DirichletspectrumMinkowskiL1normHausdorffdimensioncontinuedfractionsDiophantineapproximationlevelsets
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 paper proves that the one-dimensional Dirichlet spectrum in the L1 norm, denoted D^[1], equals the closed interval [1/2, 1]. This statement is equivalent to the Minkowski spectrum M equaling the interval [1/4, 1/2]. Constructions based on continued fractions are used to realize every value in these intervals as an approximation constant for some irrational number. The authors further determine the Hausdorff dimensions of the level sets of the Minkowski constant, showing they exceed 1/2 except at the endpoint value 1/4.

What carries the argument

The Minkowski spectrum M, the set of all attainable values of the Minkowski constant m(α) associated with the Minkowski diagonal continued fraction.

What would settle it

An explicit irrational number whose L1 Dirichlet constant lies outside [1/2, 1] or whose Minkowski constant lies outside [1/4, 1/2], or a level set whose Hausdorff dimension differs from the stated values.

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Extended reading notes

Core claim

We prove that the one-dimensional Dirichlet spectrum with respect to approximation in L1 norm D^[1] satisfies D^[1] = [1/2, 1]. This is equivalent to the Minkowski spectrum M satisfying M = [1/4, 1/2]. Further, we show that level sets Θ_m = {α ∈ (0,1) otin Q : m(α) = m} have Hausdorff dimension strictly greater than 1/2 for any m ∈ (1/4, 1/2], while dim_H Θ_{1/4} = 1/2.

Load-bearing premise

Continued fraction expansions can be constructed to realize any desired value of the approximation constant inside the claimed interval.

Editorial extensions

If this is right

  • Every real number in the interval [1/2, 1] occurs as a Dirichlet constant for L1 approximation of some irrational.
  • The Minkowski spectrum includes every value in the interval [1/4, 1/2].
  • For each m in (1/4, 1/2], the level set of irrationals with Minkowski constant exactly m has Hausdorff dimension greater than 1/2.
  • The level set of irrationals with Minkowski constant exactly 1/4 has Hausdorff dimension exactly 1/2.

Reading between the lines

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

  • Analogous constructions may determine the corresponding spectra for approximation in other norms or in higher dimensions.
  • The dimension gap at the lower endpoint suggests that constants near 1/4 are attained on comparatively smaller sets in the space of irrationals.
  • The equivalence between the two spectra may allow transfer of dimension results between different Diophantine approximation settings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 0 minor

Summary. The paper proves that the one-dimensional Dirichlet spectrum with respect to L1-norm approximation satisfies D^[1] = [1/2, 1]. This is shown to be equivalent to the Minkowski spectrum M satisfying M = [1/4, 1/2]. The authors further establish that the level sets Θ_m = {α ∈ (0,1) ot∈ Q : m(α) = m} have Hausdorff dimension strictly greater than 1/2 for m ∈ (1/4, 1/2] and exactly 1/2 when m = 1/4, via explicit recursive constructions of continued-fraction partial quotients together with pressure-function calculations and the mass-distribution principle.

Significance. If the central claims hold, the result completely determines both spectra in the L1 setting and supplies explicit constructions realizing every value in the interval together with sharp dimension estimates on the level sets. These features (parameter-free recursive constructions for partial quotients and direct application of the mass-distribution principle) constitute a concrete advance in metric Diophantine approximation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation to accept. The report accurately summarizes the main results on the L1 Dirichlet and Minkowski spectra together with the dimension estimates for the level sets Θ_m.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper establishes the spectrum equality via explicit recursive constructions of continued-fraction partial quotients that realize every m in [1/4,1/2] together with pressure-function estimates showing dim_H Θ_m > 1/2 for m > 1/4 and dim_H Θ_{1/4} = 1/2. These steps rest on the standard theory of infinite continued fractions and the mass-distribution principle; no equation reduces by construction to a fitted input, no load-bearing premise is justified solely by self-citation, and the equivalence between D^[1] and M is derived from the given definitions without circular renaming or ansatz smuggling.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The abstract presents a proof without introducing new free parameters or invented entities; it relies on standard background results in metric number theory.

assumptions (1)
  • standard math Standard properties of continued fractions and Hausdorff dimension calculations in Diophantine approximation
    The spectrum and level-set results presuppose these background facts.

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Cite this review

Pith. "Pith review of The Dirichlet spectrum with respect to $L_1$ norm is $\left[\frac12,1\right]$." pith.science (2026). https://pith.science/paper/IRAUOVRU

@misc{pith2026260608865,
  author       = {Pith},
  title        = {Pith review of: The Dirichlet spectrum with respect to $L_1$ norm is $\left[\frac12,1\right]$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRAUOVRU}},
  note         = {Machine review of arXiv:2606.08865}
}
abstract

We prove that the one-dimensional Dirichlet spectrum with respect to approximation in $L_1$ norm $\mathbb{D}^{[1]}$ satisfies $$ \mathbb{D}^{[1]}=\left[\frac12,1\right]. $$ This is equivalent to the fact that the Minkowski spectrum $\mathbb M$, associated with the Minkowski diagonal continued fraction, satisfies $$ \mathbb M=\left[\frac14,\frac12\right]. $$ Further, we show that level sets $$ \Theta_m=\{\alpha\in(0,1)\setminus\mathbb Q:\mathfrak m(\alpha)=m\}, $$ where $\mathfrak{m}(\alpha)$ is the Minkowski constant of $\alpha$, have Hausdorff dimension strictly greater than $1/2$ for any $m\in(1/4,1/2]$, while $\dim_H \Theta_{1/4}=\frac{1}{2}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

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