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arxiv: 2606.04371 · v1 · pith:ED56GRU3new · submitted 2026-06-03 · 🧮 math.DS · math.NT· math.SP

Primitive inhomogeneous approximation for fixed non-singular frequencies

Xueyin Wang This is my paper

Pith reviewed 2026-06-28 04:31 UTC · model grok-4.3

classification 🧮 math.DS math.NTmath.SP
keywords Diophantine approximationinhomogeneous approximationprimitive approximationnon-singular frequenciesLebesgue measuresimultaneous approximationdynamical systems
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The pith

For fixed non-singular frequencies in an explicit full-measure class, high-dimensional primitive inhomogeneous Diophantine approximation holds.

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

The paper establishes a primitive inhomogeneous Diophantine approximation result that applies in high dimensions when the simultaneous frequencies are fixed and non-singular. The result covers an explicit class of such frequencies that has full Lebesgue measure. A sympathetic reader would care because the statement gives a measure-theoretic guarantee for approximation properties without requiring the frequencies to vary or satisfy additional generic conditions beyond the stated class.

Core claim

For fixed non-singular simultaneous frequencies that lie in an explicit class of full Lebesgue measure, the high-dimensional primitive inhomogeneous Diophantine approximation result holds.

What carries the argument

The explicit full Lebesgue measure class of non-singular frequencies, on which the primitive inhomogeneous approximation statement is proved.

If this is right

  • The approximation statement applies simultaneously in several dimensions for fixed frequencies.
  • The result is restricted to the explicit non-singular class rather than all frequencies.
  • Full Lebesgue measure of the class ensures the property holds for almost every choice of frequencies.
  • The inhomogeneous setting is handled directly without reduction to the homogeneous case.

Where Pith is reading between the lines

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

  • The explicit class description may permit direct verification for concrete frequency vectors arising in applications.
  • The fixed-frequency setting could connect to questions of uniform distribution along orbits in the associated dynamical system.

Load-bearing premise

The frequencies must be non-singular and belong to the explicit full-measure class defined in the paper.

What would settle it

An explicit frequency inside the claimed class for which no primitive inhomogeneous approximation with the stated properties exists would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.04371 by Xueyin Wang.

Figure 1
Figure 1. Figure 1: illustrates an interior target, a target intersecting one side of the boundary, and a target intersecting two sides near a corner. 0 1 1 x1 x2 [0, 1)2 {n1α} 2rj {n2α} {n3α} [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

We prove a high-dimensional primitive inhomogeneous Diophantine approximation result for fixed non-singular simultaneous frequencies. The frequency class is explicit and has full Lebesgue measure.

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

0 major / 1 minor

Summary. The manuscript proves a high-dimensional primitive inhomogeneous Diophantine approximation result for fixed non-singular simultaneous frequencies. The frequency class is explicit and has full Lebesgue measure.

Significance. If the central proof holds, the result would advance Diophantine approximation theory by providing an explicit full-measure class of non-singular frequencies for which primitive inhomogeneous approximations are guaranteed in high dimensions, with potential applications to dynamical systems and uniform distribution.

minor comments (1)
  1. The abstract states the main theorem but does not indicate the specific sections or equations where the proof of the full-measure property or the handling of the non-singularity condition is carried out; adding explicit cross-references would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for recognizing its potential significance in advancing Diophantine approximation theory for an explicit full-measure class of non-singular frequencies. The recommendation is listed as uncertain, but no specific major comments were provided. We remain available to address any questions regarding the central proof or other aspects of the work.

Circularity Check

0 steps flagged

No circularity: direct proof of explicit full-measure result

full rationale

The manuscript states a theorem proving a high-dimensional primitive inhomogeneous Diophantine approximation result restricted to an explicitly defined full-measure class of non-singular simultaneous frequencies. No equations, definitions, or citations reduce the claimed statement to a fitted parameter, self-referential definition, or prior self-citation chain. The derivation chain is a standard mathematical proof whose scope is delimited by the stated class; the result is not forced by construction from its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.1-grok · 5528 in / 920 out tokens · 27938 ms · 2026-06-28T04:31:25.808749+00:00 · methodology

discussion (0)

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Reference graph

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