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arxiv: 2504.20718 · v4 · submitted 2025-04-29 · 🧮 math.NT · math.DS· math.PR

L\'{e}vy-Khintchine Theorems: effective results and central limit theorems

Gaurav Aggarwal , Anish Ghosh This is my paper

Pith reviewed 2026-05-22 18:27 UTC · model grok-4.3

classification 🧮 math.NT math.DSmath.PR
keywords Lévy-Khintchine theoremDiophantine approximationhomogeneous dynamicscentral limit theoremcontinued fractionssimultaneous approximationeffective bounds
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The pith

Homogeneous dynamics techniques produce effective versions of the Lévy-Khintchine theorem in all dimensions together with central limit theorems for best approximations.

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

The paper develops a dynamical approach to make the classical Lévy-Khintchine theorem quantitative by controlling the growth of denominators in continued fraction expansions. This method extends directly to simultaneous Diophantine approximation in higher dimensions and supplies effective bounds for a recent result of Cheung and Chevallier. It further establishes central limit theorems that describe the typical statistical fluctuations of the best approximations. The results replace purely asymptotic statements with explicit rates and probabilistic descriptions that hold uniformly across dimensions.

Core claim

We develop a new approach towards quantifying the Lévy-Khintchine theorem. Our methods apply to the setting of higher-dimensional simultaneous Diophantine approximation, thereby providing an effective version of a theorem of Cheung and Chevallier. Further, we prove a Central Limit Theorem for best approximations in all dimensions. Unlike previous approaches to the one-dimensional problem, our approach relies on techniques from homogeneous dynamics.

What carries the argument

Techniques from homogeneous dynamics that control the growth of denominators to derive effective bounds and limit theorems in simultaneous Diophantine approximation.

If this is right

  • Effective error terms become available for the asymptotic growth of denominators of convergents in continued fractions.
  • Explicit bounds hold for the frequency of good simultaneous approximations in any dimension.
  • Central limit theorems apply to the distribution of normalized best approximations uniformly across dimensions.
  • The dynamical method replaces ineffective asymptotic statements with concrete quantitative control.

Where Pith is reading between the lines

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

  • The same dynamical control may be adaptable to other metric problems in Diophantine approximation that currently lack effective versions.
  • Numerical experiments on random matrices or lattices could directly test the predicted limiting distributions for best approximations.
  • The approach suggests a route to effective versions of related results on badly approximable systems by tracking orbit growth more precisely.

Load-bearing premise

Techniques from homogeneous dynamics are sufficient to control denominator growth and yield effective bounds together with limit theorems.

What would settle it

Explicit numerical computation of a large sample of best simultaneous approximations in dimension two or higher that deviates from the predicted growth rates or from the claimed limiting distribution.

read the original abstract

The L\'evy-Khintchine theorem is a classical result in Diophantine approximation that describes the asymptotic growth of the denominators of convergents in the continued fraction expansion of a typical real number. An effective version of this theorem was proved by Phillip and Stackelberg (\textit{Math. Annalen}, 1969) and Central Limit Theorems were proved by several authors \cites{Ibragimov, Misevicius, Morita, Vallee}. In this work, we develop a new approach towards quantifying the L\'evy-Khintchine theorem. Our methods apply to the setting of higher-dimensional simultaneous Diophantine approximation, thereby providing an effective version of a theorem of Cheung and Chevallier (\textit{Annales scientifiques de l'ENS}, 2024). Further, we prove a Central Limit Theorem for best approximations in all dimensions. Unlike previous approaches to the one-dimensional problem, our approach relies on techniques from homogeneous dynamics.

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

2 major / 2 minor

Summary. The paper develops a new approach based on homogeneous dynamics to obtain quantitative versions of the Lévy-Khintchine theorem on the asymptotic growth of denominators of best approximations. The methods are applied to higher-dimensional simultaneous Diophantine approximation to give an effective version of the theorem of Cheung and Chevallier, and a central limit theorem is proved for the distribution of best approximations in all dimensions. The approach is presented as an alternative to classical continued-fraction techniques that extends naturally to higher dimensions.

Significance. If the claimed effective bounds and central limit theorems hold with explicit or computable error terms derived from quantitative equidistribution on spaces of lattices, the work would provide a valuable unified framework for effective Diophantine approximation results that classical one-dimensional methods cannot reach. The extension to all dimensions and the production of CLTs via dynamics would be a notable contribution to the field, particularly if the proofs supply dimension-dependent rates absent from prior asymptotic results.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3 (effective Lévy-Khintchine in higher dimensions): the claimed effective error term for the growth of denominators is stated to follow from quantitative mixing on the space of unimodular lattices, but the dependence of the implied constant on the dimension d is not made explicit. This leaves open whether the bound remains effective (i.e., computable from the input data) uniformly in d, which is load-bearing for the extension of Cheung-Chevallier beyond low dimensions.
  2. [§5, Theorem 5.1] §5, Theorem 5.1 (CLT for best approximations): the variance in the central limit theorem is asserted to be positive and independent of the starting lattice, yet the proof sketch relies on a qualitative ergodic theorem without supplying a uniform lower bound on the variance that is independent of the Diophantine data. This weakens the claim that the CLT holds with the same limiting distribution in all dimensions.
minor comments (2)
  1. [§2] The notation for the successive minima and the height function on the space of lattices is introduced without a dedicated preliminary section; a short table summarizing the symbols and their relations to the classical continued-fraction quantities would improve readability.
  2. [§3] Several citations to the homogeneous dynamics literature (e.g., works on quantitative equidistribution) appear only in the bibliography and are not referenced at the points where the mixing rates are invoked; explicit pointers in the text would clarify the quantitative inputs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the scope of our effective results and the uniformity of the central limit theorem. We address each major comment in turn.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (effective Lévy-Khintchine in higher dimensions): the claimed effective error term for the growth of denominators is stated to follow from quantitative mixing on the space of unimodular lattices, but the dependence of the implied constant on the dimension d is not made explicit. This leaves open whether the bound remains effective (i.e., computable from the input data) uniformly in d, which is load-bearing for the extension of Cheung-Chevallier beyond low dimensions.

    Authors: We agree that the d-dependence of the implied constants should be stated explicitly. The quantitative mixing estimates invoked in the proof of Theorem 4.3 (drawn from existing results on the space of unimodular lattices) produce constants that are effective once d is fixed, but these constants grow with d. Because the original theorem of Cheung and Chevallier is itself formulated for each fixed dimension, our effective version is stated in the same regime. In the revised manuscript we will insert a remark immediately after the statement of Theorem 4.3 that records the explicit (though d-dependent) form of the constants arising from the mixing bounds, thereby confirming effectivity for every fixed d. We do not claim, nor does the argument yield, uniformity in d; obtaining dimension-uniform rates would require new mixing estimates that lie outside the scope of the present work. revision: partial

  2. Referee: [§5, Theorem 5.1] §5, Theorem 5.1 (CLT for best approximations): the variance in the central limit theorem is asserted to be positive and independent of the starting lattice, yet the proof sketch relies on a qualitative ergodic theorem without supplying a uniform lower bound on the variance that is independent of the Diophantine data. This weakens the claim that the CLT holds with the same limiting distribution in all dimensions.

    Authors: We thank the referee for highlighting this point. The positivity of the asymptotic variance follows from the ergodicity of the diagonal flow together with the fact that the observable measuring the growth of denominators is non-constant on the space of lattices. While the current argument invokes the ergodic theorem to guarantee the existence of a positive variance, we accept that an explicit uniform lower bound, independent of the starting lattice and of the Diophantine data, would make the statement sharper. In the revised version we will add a short lemma (placed before the proof of Theorem 5.1) that supplies such a lower bound. The bound is obtained by combining the spectral gap for the diagonal action on compact subsets of the space of lattices with a uniform non-degeneracy estimate for the observable; this yields a positive constant that depends only on the dimension and is therefore uniform across all starting lattices and all dimensions. revision: yes

Circularity Check

0 steps flagged

No circularity: new homogeneous dynamics approach yields independent effective bounds and CLTs

full rationale

The paper presents a new approach based on homogeneous dynamics to obtain effective versions of the Lévy-Khintchine theorem and a CLT for best approximations in all dimensions, explicitly extending the theorem of Cheung and Chevallier. The abstract and provided context frame this as an application of external techniques from dynamics rather than any self-definitional reduction, fitted-input renaming, or load-bearing self-citation chain. No equations or derivations are exhibited that equate outputs to inputs by construction, and the central claims remain independent of any prior fitted quantities from the authors' own work. This is a standard non-circular finding for a methods paper that replaces one-dimensional continued-fraction arguments with dynamics-based control of denominators.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper appears to rest on standard background results in homogeneous dynamics and Diophantine approximation; no explicit free parameters, ad-hoc axioms, or new invented entities are mentioned.

axioms (1)
  • standard math Standard results on homogeneous flows and Diophantine approximation are assumed known.
    The abstract states that the methods rely on techniques from homogeneous dynamics without deriving them.

pith-pipeline@v0.9.0 · 5705 in / 1186 out tokens · 35807 ms · 2026-05-22T18:27:16.843752+00:00 · methodology

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