Fekete's lemma in Banach spaces

Aleksei Kulikov; Feng Shao

Fekete's lemma in Banach spaces

T0 review · 0 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that in every uniformly convex Banach space, sequences with subadditive vector norms have convergent averages, and it maps precisely how far the hypothesis can be relaxed.

desk verdict The main theorem is sound and genuinely new: vector-valued Fekete in uniformly convex Banach spaces, with the proof checking out; the flagged Section 3 concern is a false alarm. read the letter →

arxiv 2411.17380 v1 pith:HSGECNCO submitted 2024-11-26 math.FA math.MG

Aleksei Kulikov , Feng Shao This is my paper · ORCID

classification math.FAmath.MG MSC 46B20

keywords Fekete'slemmasubadditivesequencesBanachspacesuniformconvexityconvexvector-valueddeBruijn–Erdősconvergenceofaverages

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 Fekete's lemma says that a scalar sequence with subadditive real values has a limit after division by $n$. This paper proves the vector-valued analogue: in a uniformly convex Banach space, any sequence $v_n$ satisfying $\|v_{n+m}\|\le \|v_n+v_m\|$ for all $n,m$ admits the limit $\lim_{n\to\infty} v_n/n$ in the space. The proof isolates a geometric inequality (Lemma 2.1) that turns angular separation into a strict norm contraction, and iterates it along the index set. The paper also shows the theorem is sharp in a precise sense: convexity alone does not force convergence, uniform convexity is not necessary, and a criterion (Theorem 5.5) describes exactly when the limit exists in a convex space.

What carries the argument

The load-bearing mechanism is Lemma 2.1, a quantitative geometric inequality: in a uniformly convex Banach space, if two non-zero vectors $u,v$ have normalized directions separated by at least $\varepsilon$ and satisfy $\|v\|\le 2\|u\|$, then $\|u+v\|\le \|u\|+\gamma\|v\|$ with $\gamma=\gamma(\varepsilon)<1$. This converts angular separation into a multiplicative loss in norm. The proof applies this loss repeatedly through the iterated inequality (2.2), $\|v_{rn+m}\|\le \|v_m\|+r\gamma\|v_n\|$, valid for every $r$ up to the integer part of $m/n$, and combines it with the scalar Fekete lemma on the sequence $\|v_n\|$ to extract a contradiction if the normalized vectors fail to be Cauchy.

What would settle it

Find a sequence $\{v_n\}$ in a Hilbert space (the simplest uniformly convex space) satisfying $\|v_{n+m}\|\le \|v_n+v_m\|$ for all $n,m$ but with $\lim v_n/n$ non-existent; Theorem 1.1 asserts no such sequence exists, so any explicit construction would refute it. The paper's own counterexamples all rely on non-Hilbert geometry, making $\ell^2$ the sharp test case.

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Extended reading notes

Core claim

The central discovery is that the scalar Fekete phenomenon—subadditivity forcing $a_n/n$ to converge—persists for vector sequences precisely when the geometry of the ambient space is strong enough to make normalized vectors contract under addition. Theorem 1.1 states that in a uniformly convex Banach space, the hypothesis $\|v_{n+m}\|\le \|v_n+v_m\|$ for all $n,m$ implies the existence of $\lim_{n\to\infty} v_n/n$. The proof shows the normalized sequence $v_n/\|v_n\|$ is Cauchy: first for comparable indices, then for arbitrary indices by an iteration that accumulates a fixed geometric loss. The paper's examples delimit the result: a convex but non-uniformly-convex space is constructed where convergence fails, a non-uniformly-convex space is constructed where it holds, and Theorem 5.5 gives a necessary and sufficient condition on the normalized vectors in convex spaces.

Load-bearing premise

The proof requires the subadditivity inequality $\|v_{n+m}\|\le \|v_n+v_m\|$ to hold for every pair of indices $n,m$; weakening it to nearby pairs only, as in the de Bruijn–Erdős variant, lets the conclusion fail already in $\mathbb{R}^2$ (Section 3).

Editorial extensions

If this is right

In every uniformly convex Banach space—including all Hilbert spaces and $L^p$ spaces with $1<p<\infty$—any sequence satisfying $\|v_{n+m}\|\le \|v_n+v_m\|$ for all $n,m$ has a well-defined limit $\lim v_n/n$ in the space.
For finite-dimensional spaces, the Fekete property is equivalent to convexity (Corollary 1.2): a finite-dimensional Banach space enjoys the conclusion for all admissible sequences iff it is convex.
The nearby-pairs de Bruijn–Erdős condition $\frac12 n\le m\le 2n$ suffices in $\mathbb{R}$ but not in $\mathbb{R}^2$ with the Euclidean norm; the paper builds a sequence satisfying the nearby-pairs inequality whose averages do not converge.
In a convex Banach space, for a sequence with $\lim \|v_n\|/n>0$, the limit $\lim v_n/n$ exists if and only if $\{v_n/\|v_n\|\}$ is a uniformly convex subset (Theorem 5.5).
If some vector $v_i=0$, then $v_{ki}=0$ for all $k$, the norm averages tend to $0$, and the limit $\lim v_n/n$ is $0$ (Corollary 5.2).

Reading between the lines

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

The contraction argument in Lemma 2.1 should carry over to any metric setting with a modulus of convexity—geodesic spaces, CAT(0) spaces, or uniformly convex metric spaces—suggesting a Fekete-type theorem beyond Banach spaces.
The superlinear counterexample in $\mathbb{R}^2$ suggests a quantitative question the authors leave open: the minimal growth of the admissible ratio $f(n)$ in Question 3.2 is likely tied to the modulus of convexity of the space, which could be tested dimension by dimension.
Theorem 5.5 recasts the convergence of $v_n/n$ as a purely geometric property of the set of directions, so convergence could be verified or falsified computationally by checking uniform convexity of the normalized sequence in concrete convex spaces.

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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

The paper does what it says: proves the vector-valued Fekete lemma in uniformly convex Banach spaces, and the proof is coherent. The reader flagged a possible indexing slip in Section 3, but I checked it and it is not a slip: because m >= n, n+m >= 2n, so the ln(2n+1) lower bound is legitimate. The central argument (Lemma 2.1 plus the two-step Cauchy estimate) is sound, and the iterative bound (2.2) holds at every induction step. This is a real extension, not a repackaging: the classical references are scalar, and the MathOverflow question only posed the finite-dimensional Hilbert case.

What is genuinely new: the geometric Lemma 2.1, the sharpness examples in Section 3 showing the de Bruijn–Erdos local condition fails in R^2, and the Section 5 criterion. The examples in Section 4 are also informative, cleanly separating convexity from uniform convexity. The paper is honest about provenance and open questions.

Soft spots: Section 4 leans on 'we leave the details to the interested reader' for the non-uniformly-convex example and the non-complete space example. These are routine verifications, but a referee will want them written out or at least sketched. Section 5 is a bit quick: the criterion proof is essentially a repackaging of the main argument plus Proposition 5.3, and that is fine, but it deserved a few more sentences. None of this touches Theorem 1.1, which holds up.

The citation pattern is fine: the classical papers are cited, the MathOverflow exchange is acknowledged, and the de Bruijn–Erdos result is properly attributed. No circularity, no fitted parameters.

Who this is for: functional analysts and anyone using subadditivity arguments in Banach spaces. The main theorem will be cited. It deserves a serious referee; the referee should ask for the Section 4 details to be filled in but should not reject the paper over them.

Recommendation: send it to review.

Referee Report

0 major / 5 minor

Summary. The paper proves a vector-valued version of Fekete's subadditive lemma: in a uniformly convex Banach space X, any sequence {v_n} satisfying ||v_{n+m}|| ≤ ||v_n + v_m|| for all n,m has the property that v_n/n converges in X. The proof combines the classical scalar Fekete lemma (for the norms) with a geometric lemma (Lemma 2.1) that controls the angle between two vectors from the subadditivity-type inequality, followed by a two-scale Cauchy argument: a local ratio bound (Proposition 2.4) and an iteration argument for large ratios (display (2.2)). The paper also contains a real-line local version (Proposition 3.1), a planar counterexample showing the all-pairs condition is needed, Banach-space examples showing convexity is not sufficient and uniform convexity is not necessary for the conclusion, and a criterion (Theorem 5.5) for convergence in a general convex Banach space in terms of uniform convexity of the normalized sequence.

Significance. If correct, the main theorem is a genuine and natural extension of Fekete's lemma from scalar sequences to sequences in uniformly convex Banach spaces. The proof is elementary, self-contained, and rests on a clean geometric lemma that is likely to be useful elsewhere. The paper also crisply delineates the boundary of the result: Section 3 shows the all-pairs condition cannot be weakened to the de Bruijn--Erdős local condition, and Section 4 shows that convexity alone is insufficient while uniform convexity is not necessary. The positive credit due to the authors includes the clear two-step Cauchy argument, the explicit planar counterexample, and the acknowledgment of the MathOverflow provenance of the problem and of the contributions of F. Petrov and D. Wei. The main theorem appears sound; I checked the index hypotheses in the iteration step and the algebra in Proposition 2.4 and the final contradiction.

minor comments (5)

[Section 4, second example] The proof that Fekete's lemma holds in the constructed space X is only sketched. In particular, the sentence 'We leave the details of the computations to the interested reader' covers the crucial step of passing from the convergence of w_n/n in ℓ2 to the coordinatewise convergence of u_{n,k}/n for each k. The displayed inequality before that step contains an additional term Σ_{l≠k}(w_{n,l}+w_{m,l})² that is not present in Theorem 1.1, so the reduction to the uniform-convexity argument for a fixed pair coordinate is not automatic. Since this example supports the paper's claim that uniform convexity is not necessary, the authors should either supply the missing estimates or clearly indicate that this part is a sketch.
[Section 4, incomplete normed space example] The inequality ||v_{n+m}|| ≤ ||v_n + v_m|| is asserted without proof. It does follow from the componentwise comparison c_{n+m} ≤ c_n + c_m and the monotonicity of c_n/n, but this should be stated explicitly rather than left to the reader.
[Section 3, proof of Proposition 3.1] The bound |a_n + a_{n+1}| ≤ L + (2n+1)ε is used without derivation. A short explanation using (3.2) and the fact that a_n and a_{n+1} have opposite signs would make the argument substantially easier to follow.
[Throughout] There are several typographical errors: 'subadditivite' appears in the title and abstract, 'Exercise 6.23' in reference [1] should be 'Exercise 6.23', and in the proof of Theorem 1.1 'we already obtain edit' should read 'we already obtained it'.
[Section 5, Corollaries 5.2 and 5.4, Theorem 5.5] The notation v_n/||v_n|| presupposes that v_n ≠ 0. In Theorem 5.5 the assumption lim ||v_n||/n > 0 makes this harmless, but the text should explicitly say that the finitely many exceptional indices are ignored or handled separately.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is proved from the scalar Fekete lemma and a uniform-convexity lemma with no fitted parameters or load-bearing self-citation.

full rationale

The derivation is self-contained. Theorem 1.1 applies the classical scalar Fekete lemma to the numerical sequence ||v_n|| to obtain L = lim ||v_n||/n and the lower bound ||v_k|| >= kL after normalization. The remaining work is the geometric Lemma 2.1, which is proved directly from uniform convexity (with the proof credited to Fedor Petrov, not cited as an external theorem), and then applied only at indices where the hypotheses are verified: Proposition 2.4 handles n <= m <= 4n, and the induction for (2.2) checks at each step that m and rn+m lie in [N, 2m] so Proposition 2.4 and the norm bounds apply. No equation is defined in terms of the target limit, and no parameter is fitted to the conclusion. Section 3 relies on the independent de Bruijn-Erdos scalar theorem and explicit estimates; Section 4 gives direct constructions; Section 5's criterion is derived from the same geometric estimate and a compactness argument rather than assumed. The only self-reference is the MathOverflow provenance note [7], which is not used as evidence for any theorem. Therefore there is no circular step to report.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theorem is parameter-free. It relies on standard functional analysis background (scalar Fekete lemma, completeness, uniform convexity modulus) plus the stated hypotheses. The constructed examples in Sections 3 and 4 introduce hand-picked auxiliary sequences and norms, but these are not fitted parameters and do not support the central claim.

assumptions (6)

standard math Classical Fekete's lemma for real subadditive sequences: if a_{n+m} <= a_n + a_m, then a_n/n converges to inf_n a_n/n.
Used at the start of Section 2 to get L = lim ||v_n||/n and, after normalization, the lower bound ||v_n|| >= n for all n. Also used in Corollary 5.2.
domain assumption Uniform convexity provides a uniform modulus delta(epsilon) for all unit vectors with separation at least epsilon.
This is Definition 2 and the geometric input for Lemma 2.1; Remark 2.3 shows Lemma 2.1 is equivalent to uniform convexity.
domain assumption The vector condition ||v_{n+m}|| <= ||v_n + v_m|| implies the scalar sequence ||v_n|| is subadditive via the triangle inequality.
First sentence of Section 2; this conversion is how the classical Fekete lemma becomes available.
standard math X is complete, so a Cauchy sequence in X converges.
Final step of Theorem 1.1; the incomplete-space example at the end of Section 4 shows completeness is needed.
standard math de Bruijn-Erdos theorem [2, Theorem 22] for real sequences satisfying a_{n+m} <= a_n + a_m for 1/2 n <= m <= 2n.
Used in Proposition 3.1 to establish convergence of |a_n|/n.
standard math Fatou's lemma or monotone convergence is used to control infinite tails in the constructed non-uniformly convex example.
Section 4, final paragraph, to pass from coordinatewise limits to convergence in the ell-2-type norm.

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Cite this review

Pith. "Pith review of Fekete's lemma in Banach spaces." pith.science (2026). https://pith.science/paper/HSGECNCO

@misc{pith2026241117380,
  author       = {Pith},
  title        = {Pith review of: Fekete's lemma in Banach spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSGECNCO}},
  note         = {Machine review of arXiv:2411.17380}
}

abstract

For a sequence of vectors $\{v_n\}_{n\in\mathbb{N}}$ in the uniformly convex Banach space $X$ which for all $n, m\in \mathbb{N}$ satisfy $\|v_{n+m}\|\le \|v_n + v_m\|$ we show the existence of the limit $\lim_{n\to \infty} \frac{v_n}{n}$. This extends the classical Fekete's subadditivite lemma to Banach space-valued sequences.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Brezis, Functional analysis, Sobolev spaces and partial differential equations

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations . Universitext, Springer, New York, 2011. xiv+599 pp

work page 2011

[2] [2]

N. G. de Bruijn and P. Erd\"os, Some linear and some quadratic recursion formulas. II. Indag. Math. , 14 (1952), 152–163

work page 1952

[3] [3]

Fekete, \"Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten

M. Fekete, \"Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. , 17 (1923), 228–249

work page 1923

[4] [4]

Hille and R

E. Hille and R. S. Phillips, Functional Analysis And Semi-Groups, rev. ed . Amer. Math. Soc. Colloq. Publ., 31 . American Mathematical Society, Providence, RI, 1957. xii+808 pp

work page 1957

[5] [5]

A. Kulikov (https://mathoverflow.net/users/104330/aleksei-kulikov), High dimensional Fekete's subadditive lemma: does | x_ n+m | | x_n+ x_m| imply the convergence of \ x_n/n\ ?, URL (version: 2023-12-05): https://mathoverflow.net/q/459785

work page 2023

[6] [6]

G. S. Lueker, Improved bounds on the average length of longest common subsequences. J. ACM , 56 (2009), Art. 17, 38 pp

work page 2009

[7] [7]

I. Z. Ruzsa and Z. F\"uredi, Nearly subadditive sequences. Acta Math. Hungar. , 161 (2020), 401–411

work page 2020

[8] [8]

Shao (https://mathoverflow.net/users/141451/feng) and A

F. Shao (https://mathoverflow.net/users/141451/feng) and A. Kulikov (https://mathoverflow.net/users/104330/aleksei-kulikov), High dimensional Fekete's subadditive lemma: does | x_ n+m | | x_n+ x_m| imply the convergence of \ x_n/n\ ?, URL (version: 2023-12-05): https://mathoverflow.net/q/459773

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