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

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Pith's one-line read Any integrable function that tiles the p-adic field by translations is uniformly locally constant.

desk verdict Theorem 1.1 is probably true and worth fixing, but as written the proof skips a load-bearing distribution-theoretic identity and one direction of the Fourier-duality statement it relies on. read the letter →

arxiv 2412.03834 v2 pith:XPI7W3SJ submitted 2024-12-05 math.CA

Shilei Fan This is my paper · ORCID
classification math.CA MSC 43A70
keywords periodictranslationtilespectralsetp-adicfielduniformlylocallyconstantuniformpartitionofunityFugledeconjecturep-homogeneoustree
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 proves that any integrable function on the field $\mathbb{Q}_p$ of $p$-adic numbers that tiles the field by translations is rigid: if $\sum_{t\in T} v_t f(x-t)=w$ almost everywhere for a locally finite translation set $T$ and finitely many integer weights $v_t$, then $f$ is uniformly locally constant. That means there is a fixed scale $p^{-n}$ such that $f$ does not change when shifted by any element of $p^{-n}\mathbb{Z}_p$; in particular $f$ is periodic. The same rigidity answers, for $\mathbb{Q}_p$, a question of Leptin and Müller about uniform partitions of unity: a discrete set works precisely when its intersection counts with balls of some sufficiently large radius are independent of the ball's center. The theorem is then applied to $\mathbb{Q}_p \times \mathbb{Z}/2\mathbb{Z}$, where it yields a complete structure theory for tiles and shows that every tile is a spectral set, connecting the result to the Fuglede conjecture.

What carries the argument

The load-bearing object is the Fourier transform of the discrete measure $\nu=\sum_{t\in T}v_t\delta_t$ on the $p$-adic side. Proposition 3.1 shows that the zero set of $\hat{\nu}(\xi)=\sum_t v_t\chi(\xi t)$ is a union of spheres centered at $0$ and lies inside a bounded ball whose radius is controlled by the closest pair of points in the support of $\nu$. The sphere-orbit invariance comes from Corollary 2.4: any integer relation $\sum_j \alpha_j\chi(\xi_j)=0$ among $p$-adic characters survives multiplication of all $\xi_j$ by a unit of $\mathbb{Z}_p$, a consequence of Schoenberg's cyclotomic coefficient lemma. The argument then uses the distribution-theoretic identity $\widehat{f*\nu}=\hat{f}\cdot\hat{\nu}$; boundedness of $Z_{\hat{\nu}}$ makes the support of $\hat{f}$ compact, and the duality in Proposition 2.1 converts that into uniform local constancy of $f$.

What would settle it

Find $f\in L^1(\mathbb{Q}_p)$ that is not uniformly locally constant and a locally finite set $T$ with bounded nonzero integer weights $v_t$ such that $\sum_{t\in T}v_t f(x-t)$ is constant almost everywhere; the theorem says no such pair exists. Equivalently, exhibit $(f,\nu)$ with $\hat{f}\cdot\hat{\nu}=w\delta_0$ in the distribution sense while the support of $\hat{f}$ is non-compact.

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Extended reading notes

Core claim

Theorem 1.1 is the central claim. Let $f\in L^1(\mathbb{Q}_p)$, let $V\subset \mathbb{Z}\setminus\{0\}$ be finite, and let $T\subset\mathbb{Q}_p$ be locally finite. If weights $v_t\in V$ satisfy $\sum_{t\in T} v_t f(x-t)=w$ almost everywhere for some constant $w$, then $f$ is uniformly locally constant: there exists $n\in\mathbb{Z}$ with $f(x+u)=f(x)$ almost everywhere for every $u\in B(0,p^n)$. The proof writes the equation as $f*\nu=w$ with $\nu=\sum v_t\delta_t$, passes to the Fourier side as $\hat{f}\cdot\hat{\nu}=w\delta_0$, and shows that the zero set $Z_{\hat{\nu}}$ is bounded and rotation-invariant. Since $\hat{f}$ is continuous and vanishes outside $Z_{\hat{\nu}}\cup\{0\}$, its support is compact; the known duality between compact support and uniform local constancy then forces the conclusion.

Load-bearing premise

The proof's load-bearing premise is that the tiling equation $f*\nu=w$ can legitimately be transformed into the distribution identity $\hat{f}\cdot\hat{\nu}=w\delta_0$ for every unbounded integrable $f$ and locally finite integer-weighted $\nu$; if that Fourier transfer is not well defined for some admissible pair, the argument collapses.

Editorial extensions

If this is right

  • Any $L^1$ function tiling $\mathbb{Q}_p$ with finite integer coefficients is uniformly locally constant, so below one fixed scale every tiling function is featureless; in particular it is periodic.
  • A discrete set $T$ corresponds to a uniform partition of unity on $\mathbb{Q}_p$ if and only if the counts $\#(B(x,p^n)\cap T)$ are the same for every center $x\in\mathbb{Q}_p$ for some sufficiently large $n$; this closes the Leptin–Müller problem for $\mathbb{Q}_p$.
  • Every tile of $\mathbb{Q}_p\times\mathbb{Z}/2\mathbb{Z}$ decomposes into pieces that either share a common tiling complement in $\mathbb{Q}_p$, tile $\mathbb{Q}_p$ after merging, or (for $p=2$) fit together after a half-step shift; in every case the tile is spectral.
  • Since all tiles in $\mathbb{Q}_p\times\mathbb{Z}/2\mathbb{Z}$ are spectral, this group is another infinite abelian group where the tile-implies-spectral direction of the Fuglede conjecture holds, extending the same result already known for $\mathbb{Q}_p$.

Reading between the lines

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

  • Inference: The zero-set argument is driven by the unit group $\mathbb{Z}_p^\times$ acting transitively on spheres in $\widehat{\mathbb{Q}}_p$, so the same rigidity should hold for tiling functions on any finite extension of $\mathbb{Q}_p$ or on local function fields with the same rotation symmetry; the paper itself only treats $\mathbb{Q}_p$.
  • Inference: Theorem 4.1's formula $D(\nu)=w/\int f\,dm$ provides a quantitative, checkable invariant: for any candidate tiling pair, the average number of translate points per unit volume must converge to that ratio. This could be tested on finite truncations by computing ball counts.
  • Inference: The proof yields a computable necessary condition for a finite weighted set of translates to participate in any tiling by an $L^1$ function: the zero set of its Fourier transform must be bounded. Testing this on finite character sums would give a practical obstruction independent of solving the full tiling equation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper studies tiling equations of the form sum_{t in T} v_t f(x-t) = w on the p-adic field Q_p, where f is integrable, the weights v_t are non-zero integers from a finite set, and T is locally finite. The central result (Theorem 1.1) asserts that any such f is uniformly locally constant, i.e. constant on cosets of some small subgroup. This is applied to answer the Leptin-Müller question on uniform partitions of unity in Q_p (Theorem 1.2) and to characterize tiles in the product group Q_p x Z/2Z (Theorem 1.3), with the consequence that all such tiles are spectral (Corollary 1.4). The proof of Theorem 1.1 passes from the convolution equation f * nu = w to the distribution identity bf * bnu = w delta_0, studies the zero set of bnu, and concludes compactness of supp(bf).

Significance. If the proof can be made rigorous, Theorem 1.1 is a strong and elegant rigidity statement: integrable p-adic functions that admit a weighted tiling are automatically uniformly locally constant. The result is parameter-free and is obtained by a structural Fourier-distribution argument rather than by fitting. The applications are also significant: they give a complete answer to the Leptin-Müller question in Q_p and establish Fuglede-type spectrality for a non-trivial infinite product group. However, the significance is conditional: the central Fourier-side reduction and the final regularity step both contain unproved distribution-theoretic points, and the p=2 structural classification is imported from an unpublished same-author preprint.

major comments (5)
  1. [Section 3, proof of Theorem 1.1] The passage from the pointwise a.e. equation (1.1) to the distribution identity bf * bnu = w delta_0 is not justified by the hypotheses. Absolute convergence of sum_t v_t f(x-t) pointwise a.e. does not imply that the Bruhat-Schwartz convolution f * nu is well defined: the sum sum_t |v_t| int_K |f(x-t)| dx can diverge for a locally finite T, because local finiteness gives only finite intersections T cap (K-y) for each fixed y, not a uniform bound on the overlap number as y varies. A dominated-convergence or absolute-summability lemma for the integral version of the series must be supplied, or the theorem must be stated with an additional hypothesis that makes the distribution convolution well defined. As written, Corollary 3.3 and the proof of Theorem 1.1 rest on this unproved step.
  2. [Section 3.1, Proposition 3.1(2), Case n_nu = -infinity] The final line 'Following Proposition 2.1, we conclude that f is uniformly locally constancy' uses the implication 'supp(bf) compact implies f uniformly locally constant'. Proposition 2.1 as stated gives the opposite direction: f has compact support if and only if bf is uniformly locally constant. The needed direction is standard (inverse Fourier transform of a compactly supported bounded function is locally constant on small balls), but it is not stated or proved in the paper. Please add the correct lemma and apply it explicitly.
  3. [Sections 5.2 and 6.1, Theorems 5.3, 5.4 and 1.3] The contradiction in the first case is not valid as written. Lemma 2.6 gives |x-x0|_p <= p/|xi|_p, while the text claims 'p/|xi|_p < 1/|xi|_p', which is false for p > 1. To obtain the contradiction one must choose x0 with all other points of E at p-adic distance strictly larger than p/|xi|_p; this is possible when n_nu = -infinity, but the proof must say so. The current argument does not establish boundedness of Z_bnu in this case.
  4. [Section 6.2, proof of Corollary 1.4] Theorem 1.3(2) and the spectrality construction in Section 6.2 rely on the classification of tiles in Z/p^n Z x Z/pZ taken from the author's unpublished preprint [7] (Theorems 5.3 and 5.4). Since the p=2 case is load-bearing for the final theorem and corollary, the referee needs either a proof of these structural statements in the present paper or a reference to a published, refereed version. The paper should also state explicitly which parts of [7] are being used.
  5. [Section 6.2, proof of Corollary 1.4] In the p=2 third case, the set Lambda is defined by a formula and then declared to be a spectrum of Omega without a verification of orthonormality and completeness. This is not immediate, especially because the two fibers over Q_2 interact through 2^j0. Please provide the argument, or cite with precise statement where this is proved.
minor comments (6)
  1. [Theorem 4.1(2)] Equation (1.2) writes 'forall u in B(x,p^n)' but the intended statement is 'forall u in B(0,p^n)' (compare with the definition of uniformly locally constancy in Section 2.2).
  2. [Section 4.3, Eq. (4.7)] The statement says D(nu) equals 1/integral f dm, but Proposition 4.4 proves D(nu) = w/integral f dm. The missing w is a typo in an important theorem statement.
  3. [Section 3.1, Eq. (3.3)] Equation (4.7) interchanges the sum over lambda in E with the integral over B(0,p^n). The justification is not automatic; it should be added, for example by using the bounded-overlap information from Proposition 4.3 once it has been established.
  4. [Section 4.2, proof of Proposition 4.3] In the displayed formula (3.3), the character value should be chi(-xi x) according to formula (2.7), not chi(xi x). Since the coefficients are real, the zero sets are unaffected, but for consistency the formula should be corrected.
  5. [Throughout] The notation D(0,p^k) is used but never defined; it should be defined or replaced by an explicit set such as the sphere of radius p^k.
  6. [Throughout] There are numerous English and typographical issues (e.g., 'uniformly locally constancy', 'stats' for 'states', 'tiles' used incorrectly as a verb, inconsistent spacing). These should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem follows from distribution identities and external results; self-citations do not reduce the argument to its own inputs.

full rationale

Theorem 1.1 is proved by translating the tiling equation f*nu=w into the distribution identity \hat f * \hat nu = w delta_0 via Proposition 2.9, then proving structural facts about the zero set Z_{\hat nu} in Proposition 3.1. Those facts are proved from Lemmas 2.2-2.6 and (3.3), none of which assumes the conclusion that f is uniformly locally constant. Corollary 3.3 only uses continuity of \hat f and the distribution product identity, and the final inference from compact support of \hat f to uniform local constancy of f is the standard dual form of Proposition 2.1, applied to \hat f. There is no fitted parameter that is later renamed as a prediction, and no definition is chosen in terms of the theorem's target property. The later sections rely on the author's prior work: [5] and [6] are published external results on Fuglede's conjecture and p-homogeneous trees, and [7] is a same-author preprint giving finite-group classifications. These citations are load-bearing for the finite reduction in Theorem 1.3, but they state assumptions about finite abelian groups and do not assume the present paper's conclusions; their use is a dependence on prior work rather than a circular reduction. The omitted proofs from [7] are a completeness concern, not circularity. No specific equation or definition reduces by construction to an input of the derivation, so the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants appear; the paper is entirely structural. The load-bearing external inputs are standard p-adic Fourier analysis and the finite-group tiling classifications of [6] and the same-author preprint [7]. No new particles, forces, dimensions, or other invented entities are introduced.

assumptions (5)
  • standard math Bruhat-Schwartz distribution theory on Q_p, including Fourier transform homeomorphism on D' and convolution-multiplication duality (Proposition 2.9).
    Used throughout Section 3 to convert the convolution equation f*nu=w into the product identity bf*bnu=w delta_0; standard references [1,28,30] are cited.
  • standard math Duality between compact support and uniform local constancy for Fourier transforms on Q_p, in distribution form.
    Proposition 2.1 is stated for L1 functions; the proof of Theorem 1.1 needs the distribution version to conclude f is uniformly locally constant from compact support of bf.
  • standard math Schoenberg's theorem (Lemma 2.2) on Z-module relations among p^n-th roots of unity.
    This is the key algebraic input for rotational invariance of zeros of Fourier transforms of discrete measures, used in Lemma 2.5, Lemma 2.6, and Proposition 3.1.
  • domain assumption Finite-group tiling classifications from [6, Theorem 5.2] and [7, Theorems 5.3 and 5.4].
    Theorem 1.3 reduces the p>=3 case to [7, Theorem 5.3] and the p=2 case to [7, Theorem 5.4]; [7] is an unpublished preprint by the same author that is not reproduced in this paper.
  • domain assumption Tiles in Q_p times Z/2Z have finite Haar measure, so the indicator functions 1_Omega0 and 1_Omega1 lie in L1(Q_p).
    Theorem 1.3 applies Theorem 1.1 to f = 1_Omega0 + 1_Omega1 and f = 1_Omega0 - 1_Omega1, which requires finite measure; the paper does not state this assumption explicitly, though it is standard in the tiling literature.

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Cite this review

Pith. "Pith review of Tiling the field $\mathbb{Q}_p$ of $p$-adic numbers by a function." pith.science (2026). https://pith.science/paper/XPI7W3SJ

@misc{pith2026241203834,
  author       = {Pith},
  title        = {Pith review of: Tiling the field $\mathbbQ_p$ of $p$-adic numbers by a function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XPI7W3SJ}},
  note         = {Machine review of arXiv:2412.03834}
}
abstract

This study explores the properties of the function which can tile the field $\mathbb{Q}_p$ of $p$-adic numbers by translation. It is established that functions capable of tiling $\mathbb{Q}_p$ is by translation uniformly locally constancy. As an application, in the field $\mathbb{Q}_p$, we addressed the question posed by H. Leptin and D. M\"uller, providing the necessary and sufficient conditions for a discrete set to correspond to a uniform partition of unity. The study also connects these tiling properties to the Fuglede conjecture, which states that a measurable set is a tile if and only if it is spectral. The paper concludes by characterizing the structure of tiles in \(\mathbb{Q}_p \times \mathbb{Z}/2\mathbb{Z}\), proving that they are spectral sets.

Figures

Figures reproduced from arXiv: 2412.03834 by the authors.

Figure 1
Figure 1. The set Z/3 4Z = {0, 1, 2, · · · , 80} is considered as a tree T (4) [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. For p = 3, a TI,J -form tree with n = 5, I = {0, 2, 4}, J = {1, 3}. Now, we proceed to construct a class of subtrees of T (n) . Let I be a subset of {0, 1, . . . , n − 1}, and let J be its complement. Thus, I and J form a partition of {0, 1, . . . , n − 1}, and either set may be empty. We say a subtree TC of T (n) is of TI -form if its vertices satisfy the following conditions: (1) If i ∈ I and t0t1 . . . ti−1 is gi… view at source ↗

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Reference graph

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