The existence of full dimensional KAM tori for infinite dimensional Hamiltonian systems with long range interactions

Hongzi Cong; Juan Zhao; Wanran Ding; Zhihan Zhang

REVIEW 2 major objections 5 minor 11 references

The existence of full dimensional KAM tori for infinite dimensional Hamiltonian systems with long range interactions

T0 review · 2 major / 5 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Full-dimensional KAM tori exist for infinite mechanical systems with long-range interactions, with torus radii decaying only like exp(−ln^σ |j|) for σ>2.

desk verdict Solid incremental KAM result: full-dimensional tori with super-polynomial (but sub-exponential) decay for genuine long-range mechanical systems, under a strengthened Bourgain-type condition that only guarantees uncountably many frequencies. read the letter →

arxiv 2607.10259 v1 pith:LJIKDQ6E submitted 2026-07-11 math-ph math.DSmath.MP

Hongzi Cong , Wanran Ding , Zhihan Zhang , Juan Zhao This is my paper · ORCID

classification math-phmath.DSmath.MP MSC 37K5570H08

keywords fulldimensionalKAMtoriinfiniteHamiltoniansystemslongrangeinteractionsBourgainDiophantineconditionweightednormsslowdecayradii

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

Classical KAM theory and its early infinite-dimensional extensions demanded that interactions decay rapidly in space, so that distant degrees of freedom barely affect one another. Recent work relaxed that requirement to long-range (all-to-all) interactions, but only for tori whose action radii shrink exponentially fast. This paper shows that the same long-range systems still possess full-dimensional invariant tori whose radii decay far more slowly—only like exp(−ln^σ |j|) with σ>2. The result is obtained under a strengthened Bourgain-type Diophantine condition on the frequencies and a weighted analytic norm that keeps the long-range couplings under control. A reader who cares about infinite-dimensional Hamiltonian dynamics therefore learns that full-dimensional quasiperiodic motion survives even when every particle interacts with every other and the torus itself is only moderately thin.

What carries the argument

A weighted Fourier–Taylor norm that multiplies each multi-index by an exponential factor e^{ι w ln^σ ⌊j⌋} together with a KAM iteration that solves the homological equation under the strengthened small-divisor bound (1.8); the weight absorbs the long-range interactions while the divisor condition guarantees that the generating function remains small.

What would settle it

Exhibit a concrete long-range Hamiltonian whose frequencies satisfy the stated Diophantine bound yet whose KAM iteration diverges, or prove that the set of admissible frequencies is empty for some interval [a,b]^N.

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Extended reading notes

Core claim

Under a Bourgain-type Diophantine condition that omits the maximal multi-index and a smallness assumption on a weighted Hamiltonian norm, any real-analytic long-range perturbation of an infinite-dimensional mechanical system admits a real-analytic symplectic conjugacy that reduces the Hamiltonian to a pure frequency normal form plus a remainder of order at least two in the actions; the zero-action torus is therefore invariant and its pre-image is a full-dimensional KAM torus whose radii satisfy I_j ∼ exp(−ln^σ |j|) for σ>2.

Load-bearing premise

The frequencies must obey a Diophantine lower bound strong enough that the maximal multi-index never appears, and only uncountably many such frequencies are known to exist; the argument fails for box dimension one.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper fills a concrete intermediate slot between Dolgopyat–Fayad–Paradela’s exponential-decay long-range tori and Kuksin’s still-open polynomial-decay question. The new statement is existence of full-dimensional invariant tori with radii I_j ∼ exp(−ln^σ |j|), σ>2, for infinite-dimensional mechanical systems with all-to-all interactions. That decay is slower than pure exponential and matches Cong’s recent PDE rate, but the setting is long-range mechanical rather than short-range PDE.

What they do well is write a complete, standard Newton scheme. The weighted norm (Definition 1.1) is adapted from DFP, the Poisson-bracket and flow estimates (Lemmas 2.2–2.3, Appendix) close cleanly under σ>2, and the truncation (3.16)–(3.17) plus the Iterative Lemma control the small divisors once the maximal index is dropped from the Diophantine condition. The measure argument in Section 5 follows DFP’s box-dimension strategy and correctly shows only uncountably many frequencies, not positive product measure; they flag that d=1 is out of reach. Citations are appropriate and the self-references supply background techniques, not the target claim.

The soft spots are exactly the ones the authors advertise: the Diophantine lower bound (1.8) is stronger than classical Bourgain (no maximal index, C0(d) depending on box dimension d<1), and the frequency set is uncountable rather than large-measure. That is a genuine restriction, not a hidden flaw; the iterative estimates themselves look solid under those hypotheses. No circularity, no invented constants, no load-bearing gap visible in the written argument.

This is for people already working in infinite-dimensional KAM or long-range localization. A serious referee should see it; the result is real within its stated limits and the proof is fully written. I would send it out.

Referee Report

2 major / 5 minor

Summary. The paper proves existence of full-dimensional KAM tori for infinite-dimensional mechanical systems with long-range (all-to-all) interactions of the form (1.1)–(1.5). Under a strengthened Bourgain-type Diophantine condition (Assumption 1.1, (1.8)) that omits the maximal multi-index and requires C0(d) ≥ (4d^{2}+8d+1)/(4d(1−d)) for box dimension d∈(0,1), and under smallness of the weighted norm 9R9_ι,r,h ≤ ε (Definition 1.1), Theorem 1.2 constructs a real-analytic symplectic map Φ close to the identity conjugating H to a normal form N* plus a remainder R* containing only terms of order |α|≥2 in the actions. Corollary 1.3 then yields invariant tori with radii I_j ∼ exp(−ln^σ |j|), σ>2. The argument consists of Poisson-bracket and flow estimates (Section 2, Appendix A), a rapidly convergent KAM iteration with truncation (3.16)–(3.17) and Iterative Lemma 3.1 (Sections 3–4), and a measure estimate via box dimension showing that uncountably many frequencies satisfy (1.8) (Theorem 5.1, Section 5).

Significance. The result sits between the exponential-decay tori of Dolgopyat–Fayad–Paradela and the polynomial-decay regime posed by Kuksin for PDEs, and it extends Cong’s recent PDE work (I_j ∼ exp(−ln^σ |j|), σ>2) to long-range mechanical systems. The technical contribution is the adaptation of the weighted norm and the strengthened Diophantine condition that removes the maximal index, together with a careful reduction of the infinite union over maximal indices via box dimension (Lemma 5.2). The paper supplies a complete iterative scheme, explicit estimates, and an honest statement that only uncountably many frequencies are obtained and that d=1 lies outside the argument. These are genuine advances for infinite-dimensional KAM with long-range interactions, even though the frequency set is not of positive product measure.

major comments (2)

The Diophantine condition (1.8) and the restriction d<1 are load-bearing for the whole argument (see the truncation bound (3.17) and the covering argument of Lemma 5.2). The paper correctly proves only uncountability (Theorem 5.1) and explicitly excludes d=1 (Section 1.5). This is not a hidden gap, but the abstract and introduction should state more prominently that the result is for a zero-product-measure set of frequencies; otherwise readers may over-interpret the existence claim relative to classical KAM measure statements.
In the Iterative Lemma 3.1 the frequency-shift estimate (3.14) and the inverse-function step that produces ω*_s+1 rely on the a-priori bound (3.8) remaining strictly less than 1. The constants 0.58, 0.55, 0.5 appearing in (3.14)–(3.15) and (3.20) are chosen so that the series of ε_s^{0.5} converge, but the dependence of ε* on η, σ, w, ι is left completely implicit. A short remark quantifying how small ε must be relative to η (or at least that the iteration closes for sufficiently small ε) would make the existence statement fully checkable.

minor comments (5)

Definition 1.1 of the weighted norm 9R9_ι,r,h uses the product over j∈n of e^{ι w ln^σ ⌊ j⌋}; the same product appears with different exponents in the Poisson-bracket proof (2.10)–(2.11). A one-line reminder that the factor e^{-ι w ln^σ ⌊ℓ⌋} is absorbed by the (1-ι) weight would improve readability.
In (3.16) the truncation threshold B_s = (ln ε_s^{-1})/(w δ_s) is introduced without an explicit comparison to the Diophantine product; the subsequent estimate that the product is ≪ ε_s^{-1} for σ>2 is correct but terse. Adding the intermediate bound on ∑_{j∈k} ln ⌊ j⌋ would help.
Several references appear with future arXiv numbers (DFP26, LWYZ26). If the paper is accepted, the final versions or DOIs should be inserted when available.
Typographical: “bω” is used both for the fixed Diophantine vector and for the limiting frequency; a consistent notation (e.g., ω_* versus ω̂) would avoid confusion in Sections 3–4.
The constant C(σ,w,h) in the estimate after (3.16) is never named; writing “there exists C=C(σ,w,h)>0 such that …” would match the style of Section 5.

Circularity Check

0 steps flagged · score 1.0 of 10

Standard KAM existence proof under independently stated Diophantine and smallness hypotheses; self-citations supply technique, not the target statement.

full rationale

The derivation is a classical Newton-type KAM iteration. Assumption 1.1 (Diophantine lower bound (1.8) with exponent C0(d) depending on box dimension d∈(0,1)) and the weighted-norm smallness 9R9ι,r,h≤ε are stated as hypotheses independent of the conclusion. The Poisson-bracket estimate (Lemma 2.2), flow estimate (Lemma 2.3), vector-field bound (Lemma 2.4), truncation (3.16)–(3.17), solution of the homological equation, remainder and frequency-shift controls in the Iterative Lemma, and the measure estimate (Theorem 5.1) are all derived from these hypotheses by direct estimates; none of them is obtained by fitting a free parameter to the target or by renaming a prior result of the same authors as the present theorem. Citations to Cong’s earlier PDE work [Con24, CLSY18] and to DFP26 are used only for background techniques (weighted norms, box-dimension covering) and are not load-bearing uniqueness theorems that force the present statement. The paper itself notes that the frequency set is merely uncountable and that d=1 lies outside the argument; that restriction is an openly stated limitation, not a circularity. Consequently the circularity score is 1 (minor self-citation of technique that is not load-bearing).

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central existence statement rests on a strengthened arithmetic assumption on frequencies, analyticity and smallness of the long-range perturbation in a custom weighted norm, and several technical thresholds (σ>2, d<1, C0(d)) chosen so that the KAM iteration and the box-dimension measure estimate close. No empirical free parameters appear; the invented weighted norm is a bookkeeping device rather than a physical entity.

free parameters (4)

σ > 2
Lower bound on the logarithmic decay exponent forced by the small-divisor product estimates (3.16)–(3.17); chosen so that the exponential of B_s (ln B_s)^{1−σ} remains smaller than any negative power of ε_s.
C0(d) ≥ (4d²+8d+1)/(4d(1−d))
Minimal Diophantine exponent that makes the resonant-set measure sum converge for box dimension d∈(0,1); appears in Assumption 1.1 and Theorem 5.1.
ι ∈ (0,1) and weight w > 0
Parameters of the weighted norm 9·9_ι,r,h that control long-range couplings; free within the open interval and used to absorb losses in the Poisson-bracket estimates.
η > 0 (Diophantine constant)
Overall size of the small-divisor lower bound; the measure of the resonant set scales as η^{1−d}, so η must be taken small enough for the KAM iteration to start.

assumptions (5)

domain assumption Frequencies ω satisfy the strengthened Bourgain-type Diophantine condition (1.8) for some η>0 and some box dimension d∈(0,1).
Assumption 1.1; load-bearing for every small-divisor estimate in the iteration and for the measure argument of Section 5.
domain assumption The perturbation R is real-analytic on the complex domain D(r,h) and small in the weighted norm 9R9_ι,r,h ≤ ε.
Hypothesis of Theorem 1.2; standard analytic KAM setting adapted to the long-range weighted norm of Definition 1.1.
domain assumption The interaction is of the all-to-all form P = Σ_{i<j} P_{i,j}(θ_i,θ_j,I_i,I_j) with no a-priori spatial decay.
Model (1.1); the long-range character is what forces the stronger Diophantine condition and the weighted-norm machinery.
standard math Standard Cauchy estimates and Lie-series convergence for analytic Hamiltonian flows on Banach spaces of sequences.
Used throughout Section 2 and Appendix A to bound Poisson brackets, flows, and vector fields.
ad hoc to paper Box dimension of the frequency sequence is strictly less than 1, so that the infinite union over maximal indices can be reduced to a finite-cardinality covering (Lemma 5.2).
Section 5; the restriction d<1 is essential for the measure sum to close and is explicitly noted as currently out of reach for d=1.

invented entities (1)

Weighted norm 9R9_ι,r,h (Definition 1.1)
purpose: Controls the size of long-range Fourier–Taylor coefficients by inserting exponential weights e^{ι w ln^σ ⌊j⌋} on every site that appears in a multi-index, allowing the Poisson-bracket estimates to close without spatial decay of the interactions.
Bookkeeping device adapted from DFP26; no independent physical content, but the entire iteration is stated in this norm.

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Pith. "Pith review of The existence of full dimensional KAM tori for infinite dimensional Hamiltonian systems with long range interactions." pith.science (2026). https://pith.science/paper/LJIKDQ6E

@misc{pith2026260710259,
  author       = {Pith},
  title        = {Pith review of: The existence of full dimensional KAM tori for infinite dimensional Hamiltonian systems with long range interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LJIKDQ6E}},
  note         = {Machine review of arXiv:2607.10259}
}

read the original abstract

We prove the existence of full dimensional KAM tori for infinite dimensional mechanical systems exhibiting long range interactions, under a Diophantine condition of Bourgain type [Bourgain2005JFA], in which the radius of the invariant tori satisfies a slower decay.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 1 linked inside Pith

[1]

Bourgain

J. Bourgain. On invariant tori of full dimension for 1 D periodic NLS . J. Funct. Anal. , 229(1):62--94, 2005

2005
[2]

Cong, J.J

H.Z. Cong, J.J. Liu, Y.F. Shi, and X.P. Yuan. The stability of full dimensional KAM tori for nonlinear S chr\"odinger equation. J. Differential Equations , 264(7):4504--4563, 2018

2018
[3]

H.Z. Cong. The existence of full dimensional KAM tori for nonlinear S chr\"odinger equation. Math. Ann. , 390(1):671--719, 2024

2024
[4]

Dolgopyat, B

D. Dolgopyat, B. Fayad, and J. Paradela. Kolmogorov invariant torus theorem for weakly interacting particles i: Full dimensional tori, 2026

2026
[5]

Fröhlich, T

J. Fröhlich, T. Spencer, and C.E. Wayne. Localization in disordered, nonlinear dynamical systems. J. Statist. Phys. , 42(3-4):247--274, 1986

1986
[6]

S.B. Kuksin. Fifteen years of KAM for PDE . In Geometry, topology, and mathematical physics , volume 212 of Amer. Math. Soc. Transl. Ser. 2 , pages 237--258. Amer. Math. Soc., Providence, RI, 2004

2004
[7]

X.Z. Li, Z.F. Wang, J.G. You, and Q. Zhou. Transfer operators, canonical center dynamics, and spectral applications for long-range operators. arXiv:2606.29154 , 2026

arXiv 2026
[8]

P \"o schel

J. P \"o schel. Small divisors with spatial structure in infinite-dimensional H amiltonian systems. Comm. Math. Phys. , 127(2):351--393, 1990

1990

Show all 11 references

[9]

Y.F. Shi. A multi-scale analysis proof of the power-law localization for random operators on Z ^d . J. Differential Equations , 297:201--225, 2021

2021
[10]

Y.F. Shi. Localization for almost-periodic operators with power-law long-range hopping: A N ash-- M oser iteration type reducibility approach. Comm. Math. Phys. , 402:1765--1806, 2023

2023
[11]

Shi and L

Y.F. Shi and L. Wen. Green's function estimates for quasi-periodic operators on Z ^d with power-law long-range hopping. Adv. Math. , 482:110661, 2025

2025

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.

[1] [1]

Bourgain

J. Bourgain. On invariant tori of full dimension for 1 D periodic NLS . J. Funct. Anal. , 229(1):62--94, 2005

2005

[2] [2]

Cong, J.J

H.Z. Cong, J.J. Liu, Y.F. Shi, and X.P. Yuan. The stability of full dimensional KAM tori for nonlinear S chr\"odinger equation. J. Differential Equations , 264(7):4504--4563, 2018

2018

[3] [3]

H.Z. Cong. The existence of full dimensional KAM tori for nonlinear S chr\"odinger equation. Math. Ann. , 390(1):671--719, 2024

2024

[4] [4]

Dolgopyat, B

D. Dolgopyat, B. Fayad, and J. Paradela. Kolmogorov invariant torus theorem for weakly interacting particles i: Full dimensional tori, 2026

2026

[5] [5]

Fröhlich, T

J. Fröhlich, T. Spencer, and C.E. Wayne. Localization in disordered, nonlinear dynamical systems. J. Statist. Phys. , 42(3-4):247--274, 1986

1986

[6] [6]

S.B. Kuksin. Fifteen years of KAM for PDE . In Geometry, topology, and mathematical physics , volume 212 of Amer. Math. Soc. Transl. Ser. 2 , pages 237--258. Amer. Math. Soc., Providence, RI, 2004

2004

[7] [7]

X.Z. Li, Z.F. Wang, J.G. You, and Q. Zhou. Transfer operators, canonical center dynamics, and spectral applications for long-range operators. arXiv:2606.29154 , 2026

arXiv 2026

[8] [8]

P \"o schel

J. P \"o schel. Small divisors with spatial structure in infinite-dimensional H amiltonian systems. Comm. Math. Phys. , 127(2):351--393, 1990

1990