Pith. sign in

REVIEW 3 major objections 5 minor 20 references

Exponential Riesz bases in non-Archimedean locally compact Abelian groups

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Every compact open subset of a non-Archimedean locally compact abelian group admits a Riesz basis of characters, while some bounded open sets admit none.

desk verdict Solid p-adic proofs of two genuinely new theorems, but the general-group non-existence proof needs explicit fill-ins before it is complete. read the letter →

arxiv 2505.21310 v2 pith:SEAWVKOB submitted 2025-05-27 math.CA

Aihua Fan , Shilei Fan This is my paper
classification math.CA MSC 43A75
keywords Rieszbasisexponentialnon-Archimedeangrouplocallycompactabelianp-adicnumbersopensubgroupcharactersframes
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

The paper asks which square-integrable function spaces on a non-Archimedean locally compact abelian group can be spanned by a Riesz basis of continuous characters, called exponentials. It proves that every compact open subset $\Omega$ admits such a basis $\Lambda \subset \widehat{G}$. It also proves the opposite extreme: in every non-discrete such group, some bounded open set of positive finite Haar measure admits no character Riesz basis at all, with an explicit construction in the $p$-adic numbers. Together the two results settle the compact-open case completely and transfer to non-Archimedean groups a phenomenon previously known for bounded sets in the real line.

What carries the argument

The central machinery is the coset decomposition of compact open sets by compact open subgroups together with Pontryagin duality. A compact open set $\Omega$ is a finite disjoint union of cosets $c+H$ of a compact open subgroup $H$; the quotient $\widehat{G}/H^\perp$ is discrete, and $H^\perp$ is identified with the dual of $G/H$. For existence, the paper chooses a finite set $D$ of characters making the evaluation matrix on the finite coset set $C$ invertible, then forms $\Lambda=D+L$, where $L$ is a complete set of coset representatives of $\widehat{G}/H^\perp$; orthogonality across distinct $H^\perp$-cosets yields both the frame and the Riesz-sequence estimates. For non-existence, the load-bearing object is the $B$-translation number $N_B(S)$, the largest cardinality of a set of disjoint translates of $S$ lying inside a compact open subgroup $B$; Proposition 1.4 shows that unbounded growth of $N_{B_n}(\Omega_n)$ along a nested chain $B_n$ contradicts the coefficient bounds any Riesz basis must satisfy.

What would settle it

A concrete test for the non-existence claim is to compute the translation numbers $N_{p^{m_n}\mathbb{Z}_p}(\Omega_n)$ for the example $\Omega=\bigcup_n (p^{m_n-1}+p^{m_n}\mathbb{Z}_p)$ and check whether they are unbounded as $n\to\infty$; if they are bounded, Proposition 1.4 would not apply. For the existence side, take a small compact open set such as $\mathbb{Z}_p\cup(1+p\mathbb{Z}_p)$, build $\Lambda$ by the paper's recipe, and directly verify the frame and Riesz-sequence inequalities; a failure would disprove Theorem 1.1. Separately, checking whether a non-Archimedean non-discrete group can have a compact open subgroup chain with bounded index ratios would test the structural assumption behind Theorem 1.5.

Watch

Extended reading notes

Core claim

Theorems 1.1 and 1.5 carry the paper. Theorem 1.1 states that for any compact open subset $\Omega$ of a non-Archimedean l.c.a.\ group $G$, there exists a set $\Lambda \subset \widehat{G}$ such that the characters $\{\chi_\lambda\}_{\lambda\in\Lambda}$ form a Riesz basis of $L^2(\Omega)$. Theorem 1.5 states that in every non-Archimedean, non-discrete l.c.a.\ group there is a bounded open set $\Omega$ with $0<\mu(\Omega)<+\infty$ for which no subset of characters forms a Riesz basis of $L^2(\Omega)$. The explicit $p$-adic instance is $$\$\Omega$=\bigcup_{n\ge 1}\left($p^{{m_n-1}}$+$p^{{m_n}}$\mathbb{Z}_p\right)$$ with $\limsup_n (m_{n+1}-m_n)=+\infty$, for example $m_n=n^2$. The non-existence proof rests on Proposition 1.4: if the tails of such a disjoint union admit unboundedly many disjoint translates inside nested compact open subgroups, then no character Riesz basis can exist.

Load-bearing premise

The general-group proofs assume that non-Archimedean locally compact abelian groups behave exactly like $\mathbb{Q}_p$: every compact open set is a finite union of cosets of a compact open subgroup, characters are locally constant, the stated duality identifications hold, and every non-discrete group has nested compact open subgroups whose index ratios tend to infinity.

Editorial extensions

If this is right

  • Every compact open subset of $\mathbb{Q}_p^d$, in particular every finite union of $p$-adic balls, admits a Riesz basis of characters (Corollary 1.2).
  • A Borel set $\Omega\subset\mathbb{Q}_p$ of positive finite Haar measure that $k$-tiles $\mathbb{Q}_p$ by translations has a character Riesz basis (Corollary 1.3).
  • In every non-Archimedean non-discrete l.c.a.\ group, bounded open sets can be spectrally bad: the compact-open case is the only unconditionally good case among open sets (Theorem 1.5).
  • Sets such as $\bigcup_{n\ge 1}(p^{n^2}+p^{n^2+1}\mathbb{Z}_p)$ are explicit $p$-adic counterparts of the Euclidean set with no exponential Riesz basis (Corollary 1.6).

Reading between the lines

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

  • The same translation-number criterion could produce non-existence examples in many more groups: any nested sequence of compact open subgroups with unbounded index ratios should yield a bounded open set with no exponential Riesz basis, not only the explicit $p$-adic example.
  • The paper leaves open whether every multi-tiling set in $\mathbb{Q}_p^d$ admits a character Riesz basis; a natural next step is to adapt the coset-and-finite-matrix construction to product groups through the product structure of the dual.
  • If some exotic non-Archimedean group fails the asserted subgroup-chain property, Theorem 1.5 would still hold there only under an explicit weaker condition; checking which groups admit index-unbounded chains would delineate the theorem's true scope.
  • One could compute the optimal frame bounds for small $p$-adic compact open sets to see how the constants in the construction depend on the geometry of the finite set $C$ and the scale parameter $\gamma$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies exponential (character) Riesz bases in non-Archimedean locally compact abelian groups. Theorem 1.1 asserts that every compact open subset Ω of such a group G admits a set Λ ⊂ Ĝ forming a Riesz basis of L²(Ω). Theorem 1.5 asserts that every non-discrete non-Archimedean l.c.a. group contains a bounded open set Ω of positive finite measure with no character Riesz basis, and Corollary 1.6 gives an explicit p-adic example. The proofs are developed in detail for Q_p: Theorem 3.2 proves existence for compact open subsets of Z_p via a character-matrix argument, and Theorem 4.1 proves non-existence for a class of open sets with large translation numbers. Sections 3.3 and 4.2 then claim that the same arguments extend to general non-Archimedean l.c.a. groups, with Proposition 1.4 as the general non-existence criterion.

Significance. If the claims are fully established, the results are significant: Theorem 1.1 would settle the positive side for the natural class of compact open sets in all non-Archimedean l.c.a. groups, and Theorem 1.5 would provide the first non-Archimedean analogues of the Euclidean non-existence phenomenon of Kozma, Nitzan, and Olevskii. The p-adic core of the paper is convincing: the frame estimate in Theorem 3.2 follows from invertibility of the character matrix, the Riesz-sequence estimate follows from orthogonality of characters separated by p-adic distance, and the translation argument in Theorem 4.1 is elementary and checkable. The explicit p-adic example in Corollary 1.6 is concrete and falsifiable. The main weakness is that the general-group versions of both main theorems are asserted by delegation to the p-adic case, and several load-bearing structural facts in Section 4.2 are stated without proof. These gaps appear fillable, but the manuscript as written does not yet prove the full generality of Theorems 1.1 and 1.5.

major comments (3)
  1. [§4.2, proof of Theorem 1.5] The line "Since G is non-discrete, there exists a nested compact open subgroups {B_n} such that lim #(B_n/B_{n+1})=+∞" is asserted without proof. This is a nontrivial structural fact about non-Archimedean l.c.a. groups; a proof or precise reference is needed because the construction of Ω depends entirely on it. Moreover, the next line "By (4.22), lim N_{B_n}(Ω_n)=+∞" is not a direct consequence of (4.22). Since Ω_n contains x_{n+1}+B_{n+1}, two translates Ω_n+t and Ω_n+t' can be disjoint only if t−t' avoids the coset x_{n+1}+B_{n+1} (and, for the residual part, B_{n+1}); hence the full set of coset representatives of B_n/B_{n+1} is not an admissible translation set. One needs an independent-set argument showing that a positive fraction of the cosets can be selected, which still gives divergence, but the written proof omits this step.
  2. [§4.2, proof of Proposition 1.4] The proof begins "Fix a compact open subgroup U ... Without loss of generality, we assume Ω ⊂ U". Since "bounded" is never defined for general l.c.a. groups in the paper, this reduction needs justification: one must show that any bounded open set of the displayed form lies in some compact open subgroup. Later, the lower bound is asserted for "all t ∈ U_mn"; this notation is undefined and appears to be a typo for B_n or for p^{m_n}Z_p. The passage from the pointwise lower bound on ∫_{Ω_n+t}|\tilde f_n|² to the sum over T also depends on the translates Ω_n+t being pairwise disjoint; this is part of the definition of a translation set, but the proof should state explicitly that T is chosen inside B_n and that the displayed sum is over disjoint sets, so that the inequality for ||\tilde f_n||²_{L²(U)} is fully justified.
  3. [§3.3, proof of Theorem 1.1] The general-group proof of Theorem 1.1 says that the p-adic argument carries over by the same reasoning, but the key estimates are not reproduced. The p-adic proof of Theorem 3.2 uses the explicit structure of L_γ, the invertibility of the character matrix (χ_{−d}(c)), and Parseval identities for L²(Z_p); none of these are stated in the general-group setting. In particular, analogues of the frame lower bound (3.9) and the Riesz-sequence lower bound (3.15) are needed for the system indexed by D+L, where D⊂H^⊥ and L is a coset-representative set for Ĝ/H^⊥. Please either give the full argument or provide precise references for each structural fact (annihilator identifications, orthogonality of characters across cosets of H^⊥, and the local-constancy reduction). As written, Theorem 1.1 is proved in detail only for Q_p.
minor comments (5)
  1. [Abstract and Introduction] The phrase "exponential Riesz basis" is used inconsistently; the theorems speak of "Riesz basis of characters" or "Riesz basis of exponentials". Please standardize the terminology.
  2. [§3.2, after (3.12)] The equality |ℓ+d−(ℓ′+d′)|_p = |ℓ−ℓ′|_p ≥ p^{γ+1} uses the non-Archimedean triangle inequality together with d−d′ ∈ Γ_γ, where |d−d′|_p ≤ p^γ. Adding one sentence would make the orthogonality step clearer.
  3. [§4.1, proof of Theorem 4.1] The notation N_{m_n}(Ω_n) appears at the end of the proof after N_{p^{m_n}Z_p}(Ω_n) was defined; please use a single consistent notation throughout.
  4. [§2.2, example (3)] The text "Zn_p" should read "Z_p^n"; the same superscript issue appears in several places.
  5. [§4.2, proof of Theorem 1.5] The statement "Noting that Ω_n ⊂ B_n" is true because x_{n+1} ∈ B_n \ B_{n+1}, but it should be spelled out that the whole union Ω_n is contained in B_n, which also implies that the constructed set Ω is bounded in the sense of being relatively compact (indeed contained in B_0).

Circularity Check

1 steps flagged · score 2.0 of 10

Only a minor same-author citation supports Corollary 1.3; the central existence and non-existence theorems are derived from in-paper lemmas and standard external facts, not by construction.

  1. self citation load bearing [Section 1, Corollary 1.3]
    "As a direct consequence of [8, Theorem 1.1], we obtain the following corollary. Corollary 1.3. Let Ω ⊆ Qp be a Borel set of positive and finite Haar measure. If Ω k-tiles Qp by translations, then L2(Ω) admits a Riesz basis consisting of characters in ̂Qp."

    Corollary 1.3 is not proved in the text; it is asserted to follow from [8, Theorem 1.1], an arXiv preprint by coauthor S. Fan. The implication is transferred from a same-author source rather than derived in this paper, so within this paper the support for the corollary reduces to a self-citation. This is, however, ancillary: Corollary 1.3 is not used to establish Theorem 1.1 or Theorem 1.5, and the main theorems rest on Lemma 2.1, Lemma 3.1, standard duality facts, and the in-paper translation argument.

full rationale

The central derivation chain is self-contained. Theorem 1.1 is built from Lemma 3.1 (finite-subset Riesz bases in discrete groups), the p-adic coset decomposition, and standard Pontryagin duality identifications in the general-group section; the general case is explicitly delegated to the p-adic proof, which is a completeness gap rather than a circular reduction. Proposition 1.4 and Theorem 1.5 use a translation-set counting argument modeled on Kozma-Nitzan-Olevskii [17], with the nonexistence following from N_Bn(Ω_n)→∞; no fitted parameter is renamed as a prediction. The proof of Theorem 1.5 does assert without demonstration that non-discreteness yields nested compact open subgroups with #(B_n/B_{n+1})→∞ and that (4.22) forces N_Bn(Ω_n)→∞; these are unproved structural claims, but they are not equivalences with the conclusion and are fillable, so they belong to correctness risk, not circularity. The only same-author dependence is Corollary 1.3, which cites the authors' own [8] as its entire proof; it is a minor self-citation and is not load-bearing for the paper's main claims. Accordingly the circularity score is 2, not higher.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

Pure-mathematics paper: zero free parameters, no data-fitted constants, and no invented entities beyond within-proof constructions (the sets Ω, the frequency sets Λ, and the translation number N_B(S), which is a definition rather than an entity). The central claims rest on standard harmonic-analysis background (axioms 1-3) and on one true-but-unproved profinite-group fact (axiom 4). The principal external dependence is the same-author preprint [8] for the peripheral Corollary 1.3; the main theorems do not depend on the authors' prior work.

assumptions (4)
  • standard math Pontryagin duality for l.c.a. groups: for a compact open subgroup H of G, restriction gives Ĝ/H^⊥ ≅ Ĥ, and H^⊥ is the dual of the discrete quotient G/H; characters on H forming a complete set of representatives of Ĥ ≅ Ĝ/H^⊥ give an orthogonal basis of L²(H).
    Invoked in §3.3 to transfer the p-adic construction to general groups; the p-adic case uses the same facts via L_γ as an index set for an orthogonal basis of L²(p^γZ_p).
  • domain assumption van Dantzig / local structure: every compact open subset of a totally disconnected l.c.a. group is a finite disjoint union of cosets of a single compact open subgroup, and every continuous character is locally constant on compact open subgroups.
    Used in §3.3 to write Ω = ⊔_{c∈C}(c + H), and in §4.1 and §4.2 ('local constancy of characters') to reduce a hypothetical basis Λ to the union of annihilators ∪_n B_n^⊥.
  • standard math p-adic character orthogonality (equation (2.5)): ∫_{p^{-n}Z_p} χ dµ = 0 for all n ≥ 1, equivalently ∫_{c+p^γZ_p} χ_{λ−λ'} = 0 whenever |λ−λ'|_p ≥ p^{γ+1}.
    Load-bearing for the Riesz-sequence half of Theorem 3.2: it kills all cross terms between different ℓ, ℓ′ ∈ L_γ in the expansion (3.12)-(3.13).
  • domain assumption Every non-discrete non-Archimedean l.c.a. group contains nested compact open subgroups B_0 ⊃ B_1 ⊃ ... with #(B_n/B_{n+1}) → ∞, and for Ω = ⊔(x_n+B_n) with x_n ∈ B_{n−1} \ B_n this yields N_{B_n}(Ω_n) → ∞.
    Asserted as (4.22) and used one line later in the proof of Theorem 1.5; the paper gives no proof of either statement. The first is true (infinite profinite groups have open subgroups of arbitrarily large index); the second follows by translating Ω_n by coset representatives of B_n/B_{n+1}.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Exponential Riesz bases in non-Archimedean locally compact Abelian groups." pith.science (2026). https://pith.science/paper/SEAWVKOB

@misc{pith2026250521310,
  author       = {Pith},
  title        = {Pith review of: Exponential Riesz bases in non-Archimedean locally compact Abelian groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SEAWVKOB}},
  note         = {Machine review of arXiv:2505.21310}
}
read the original abstract

This paper establishes two fundamental results on the existence of exponential Riesz basis in non-Archimedean locally compact Abelian groups: the existence of Riesz basis of exponentials for all finite unions of balls and the non-existence of such basis for some bounded sets.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [8]

    Tiling the field $\mathbb{Q}_p$ of $p$-adic numbers by a function

    S. F an, Tiling the field Qp of p-adic numbers by a function , arXiv preprint arXiv:2412.03834, (2024)

    work page Pith review arXiv 2024

  2. [1]

    Agora, J

    E. Agora, J. Antezana, and C. Cabrelli, Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups , Adv. Math., 285 (2015), pp. 454–477

    work page 2015

  3. [2]

    Bezuglaya and V

    L. Bezuglaya and V. Katsnelson, The sampling theorem for functions with limited multi- band spectrum, Z. Anal. Anwendungen, 12 (1993), pp. 511–534

    work page 1993

  4. [3]

    J. W. S. Cassels and A. Fr¨ohlich, eds., Algebraic Number Theory, Academic Press, 1967. Reprinted by Cambridge University Press

    work page 1967

  5. [4]

    Christensen, An introduction to frames and Riesz bases , Applied and Numerical Harmonic Analysis, Birkh¨ auser/Springer, [Cham], second ed., 2016

    O. Christensen, An introduction to frames and Riesz bases , Applied and Numerical Harmonic Analysis, Birkh¨ auser/Springer, [Cham], second ed., 2016

    work page 2016

  6. [5]

    Debernardi and N

    A. Debernardi and N. Lev, Riesz bases of exponentials for convex polytopes with symmetric faces, J. Eur. Math. Soc. 24 (2022), pp. 3017–3029

    work page 2022

  7. [6]

    A. F an, S. F an, and R. Shi, Compact open spectral sets in Qp, J. Funct. Anal. 271 (2016), no. 12, pp. 3628-3661

    work page 2016

  8. [7]

    A. F an, S. F an, L. Liao, and R. Shi, Fuglede’s conjecture holds in Qp, Math. Ann., 375 (2019), pp. 315–341

    work page 2019

Show all 20 references
  1. [9]

    Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J

    B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis, 16 (1974), pp. 101–121

    1974

  2. [10]

    Gao, Invariant descriptive set theory , vol

    S. Gao, Invariant descriptive set theory , vol. 293 of Pure and Applied Mathematics (Boca Raton), CRC Press, Boca Raton, FL, 2009

    2009

  3. [11]

    Grepstad and N

    S. Grepstad and N. Lev, Multi-tiling and Riesz bases , Adv. Math., 252 (2014), pp. 1–6

    2014

  4. [12]

    Hewitt and K

    E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups , vol. Band 152 of Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York-Berlin, 1970

    1970

  5. [13]

    Kohlenberg, Exact interpolation of band-limited functions , J

    A. Kohlenberg, Exact interpolation of band-limited functions , J. Appl. Phys., 24 (1953), pp. 1432–1436

    1953

  6. [14]

    M. N. Kolountzakis, Multiple lattice tiles and Riesz bases of exponentials , Proc. Amer. Math. Soc., 143 (2015), pp. 741–747. EXPONENTIAL RIESZ BASES IN NON-ARCHIMEDEAN L. C. A. GROUPS 15

    2015

  7. [15]

    Kozma and S

    G. Kozma and S. Nitzan, Combining Riesz bases , Invent. Math., 199 (2015), pp. 267–285

    2015

  8. [16]

    , Combining Riesz bases in Rd, Rev. Mat. Iberoam., 32 (2016), pp. 1393–1406

    2016

  9. [17]

    Kozma, S

    G. Kozma, S. Nitzan, and A. Olevskii, A set with no Riesz basis of exponentials , Rev. Mat. Iberoam., 39 (2023), pp. 2007–2016

    2023

  10. [18]

    Lev and M

    N. Lev and M. Matolcsi, The fuglede conjecture for convex domains is true in all dimensions , Acta Math., 228 (2022), pp. 385–420

    2022

  11. [19]

    Seip, A simple construction of exponential bases in L2 of the union of several intervals , Proc

    K. Seip, A simple construction of exponential bases in L2 of the union of several intervals , Proc. Edinburgh Math. Soc. (2), 38 (1995), pp. 171–177

    1995

  12. [20]

    V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-adic analysis and mathematical physics, vol. 1 of Series on Soviet and East European Mathematics, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. (A. H. F AN) 1. Wuhan Institute for Math & AI, Wuhan University...

    1994

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.