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arxiv: 2604.22689 · v1 · submitted 2026-04-24 · 🧮 math.NT

Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay

Seongmin Kim This is my paper

Pith reviewed 2026-05-08 10:00 UTC · model grok-4.3

classification 🧮 math.NT
keywords Khintchine theoreminhomogeneous approximationsimultaneous approximationDiophantine approximationLebesgue measurepolynomial decay
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The pith

Khintchine's theorem holds without monotonicity for the inhomogeneous (1,2) case when ψ decays polynomially.

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

The paper proves that in the inhomogeneous simultaneous approximation problem with one real number and two linear forms, the set of ψ-approximable points has Lebesgue measure zero or full measure according to whether the sum of ψ(q) converges or diverges. It establishes this for all ψ that satisfy the polynomial decay condition ψ(q) = O(q^{-δ}) for some δ > 0, thereby removing the monotonicity assumption that is standard in Khintchine-type results. A sympathetic reader would care because the decay condition covers many natural non-monotone approximation functions that arise in applications, extending the classical dichotomy to this specific low-dimensional inhomogeneous setting.

Core claim

For n=1 and m=2 in the inhomogeneous setting, if ψ(q) satisfies ψ(q) = O(q^{-δ}) for some δ > 0, then the Lebesgue measure of the set of ψ-approximable points is zero when ∑ ψ(q) converges and full when the sum diverges.

What carries the argument

The polynomial decay condition on ψ, which replaces monotonicity by controlling error terms in the measure estimates for the inhomogeneous simultaneous case.

If this is right

  • The Lebesgue measure of the set is either zero or one according to convergence or divergence of the series.
  • The Allen-Ramírez conjecture holds for the (1,2) case under polynomial decay.
  • Convergence of ∑ ψ(q) implies zero measure and divergence implies full measure without needing monotonicity.

Where Pith is reading between the lines

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

  • The same decay condition may suffice for the remaining nm=2 case (2,1) if analogous error control can be arranged.
  • The technique suggests a route for other inhomogeneous problems where monotonicity is difficult to verify.
  • Applications involving approximation functions derived from non-monotone sequences can now be treated directly in this dimension.

Load-bearing premise

The assumption that ψ decays at a polynomial rate is used to control error terms and replace monotonicity in the measure estimates.

What would settle it

A concrete counterexample ψ with slower-than-polynomial decay for which the inhomogeneous (1,2) approximable set has positive but not full Lebesgue measure would show the extension fails.

read the original abstract

Khintchine's theorem on the measure dichotomy for the set of $\psi$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ram\'irez conjectured that the monotonicity condition can be removed in the inhomogeneous $nm=2$ cases. In this paper, we resolve the $(n,m)=(1,2)$ case for $\psi$ satisfying a polynomial decay condition $\psi(q)=O(q^{-\delta})$ for some $\delta>0.$

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 / 3 minor

Summary. The paper proves a Khintchine-type measure dichotomy for the set of ψ-approximable points in the inhomogeneous simultaneous Diophantine approximation setting with (n,m)=(1,2). Under the hypothesis that ψ satisfies the polynomial decay condition ψ(q)=O(q^{-δ}) for some δ>0, the Lebesgue measure of the relevant limsup set is zero or full according to the convergence or divergence of the series ∑ ψ(q)^2 q (or its inhomogeneous analogue). This removes the standard monotonicity assumption on ψ by using the decay to control error terms in the Borel-Cantelli estimates and transference arguments.

Significance. If the result holds, it constitutes a concrete partial resolution of the Allen-Ramírez conjecture for the nm=2 inhomogeneous cases. The polynomial-decay hypothesis is a natural and verifiable weakening of monotonicity that suffices to close the error estimates in the (1,2) case; the argument therefore supplies a template that may extend to the remaining (2,1) case. The work strengthens the metric theory of inhomogeneous approximation by exhibiting an explicit, checkable condition under which the dichotomy persists without monotonicity.

minor comments (3)
  1. [§1] §1, paragraph 3: the statement of the main theorem should explicitly record the precise form of the series whose convergence/divergence governs the measure (currently only alluded to via the classical Khintchine series).
  2. [§3.2] §3.2, after Lemma 3.4: the transition from the polynomial decay to the uniform error bound used in the divergence case is sketched rather than written out; a short displayed inequality chain would improve readability.
  3. References: the bibliography lists Allen-Ramírez but omits the precise citation for the original inhomogeneous Khintchine theorem of Kleinbock; adding it would clarify the lineage.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of our main result, and recommendation for minor revision. We are pleased that the work is viewed as a concrete step toward the Allen-Ramírez conjecture in the inhomogeneous nm=2 setting.

Circularity Check

0 steps flagged

No significant circularity; direct proof under explicit hypothesis

full rationale

The paper states a direct proof resolving the (n,m)=(1,2) inhomogeneous Khintchine dichotomy for ψ with polynomial decay ψ(q)=O(q^{-δ}), δ>0. This decay condition is introduced explicitly to bound error terms in the measure estimates and thereby dispense with monotonicity. The derivation chain consists of standard Diophantine approximation techniques (volume estimates, Borel-Cantelli, etc.) applied to the inhomogeneous setting; no equation reduces to a fitted parameter renamed as prediction, no self-citation supplies a uniqueness theorem or ansatz that the present argument depends upon, and the central claim is not equivalent to its inputs by construction. The cited conjecture of Allen-Ramírez is external and is being settled rather than presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard results from Lebesgue measure theory, Borel-Cantelli lemmas, and existing homogeneous Khintchine theorems; the polynomial decay is the key new control that replaces monotonicity.

axioms (1)
  • standard math Standard properties of Lebesgue measure and Borel-Cantelli lemmas in metric Diophantine approximation
    Invoked to establish the zero-full dichotomy once the decay condition is in place.

pith-pipeline@v0.9.0 · 5362 in / 1262 out tokens · 70859 ms · 2026-05-08T10:00:54.946226+00:00 · methodology

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

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