{"id":"8dc6d476-2297-4a9e-95eb-86d490bc695d","arxiv_id":"2412.03834","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Integrable function tilings of Q_p are uniformly locally constant, yielding a complete answer to Leptin-Müller's question on Q_p and spectrality of tiles in Q_p times Z/2Z.","lead":"This paper proves that any integrable function that tiles the p-adic numbers by translations must be uniformly locally constant, extending an earlier result for indicator functions of sets. It uses this to answer a question of Leptin and Müller about uniform partitions of unity on Q_p and to show that every tile of Q_p times Z/2Z is a spectral set.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's proof hinges on the unproved distributional identity bf·bnu=wδ0; pointwise absolute convergence of the tiling sum does not by itself make the Bruhat-Schwartz convolution/product well-defined.","rationale":"The reader's weakest_assumption coincides with the main gap I identify: the passage from the pointwise convolution equation to the distributional Fourier identity bf·bν=wδ0 is asserted rather than proved for the class of data in Theorem 1.1. I do not see an internal contradiction that would force rejection; the theorem may be true, but the proof as written is incomplete at this Fourier passage. Secondary issues—the empty-T edge case, the converse direction of Proposition 2.1, and the p=2 dependence on the unpublished preprint—reinforce but do not replace this primary concern. Since the reader already assigned a conditional verdict, my read does not change the verdict.","tokens_in":17050,"tokens_out":49534,"duration_ms":532584,"concrete_test":"For φ=1_{B(0,p^m)}, compute ⟨f*ν,φ⟩ and ⟨bf·bν,φ⟩ directly from the definitions in §2.6, without invoking Proposition 2.9, for a locally finite T with clusters of size N_j→∞ at scales j→∞ and an f satisfying the pointwise equation (1.1). Check whether the k→∞ limits exist and equal w∫φ. In parallel, attempt the explicit cluster ansatz: disjoint contribution balls D_j, T_j={z_j−y_j, z_j−y_j+a_j}, and f defined on B(y_j,p^{-j}) by h_j with h_j(u)=−h_j(u−a_j), scaled so that f∈L1 and the pointwise sum converges absolutely; if such an f is not uniformly locally constant, Theorem 1.1 is false, and if no such f exists, the missing dominated-convergence lemma should be written and checked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 reduces f*nu=w to bf·bnu=wδ0 by invoking Proposition 2.9(3), but Proposition 2.9 applies only when both convolution and product are well-defined as Bruhat-Schwartz distributions. The hypotheses of Theorem 1.1 give only pointwise absolute convergence of sum_{t∈T} v_t f(x−t) to w a.e.; they do not imply that the canonical pairing with a test function φ, namely sum_t v_t ∫ f(x)φ(x+t)dx, is absolutely convergent. T is only locally finite, so it may contain clusters of arbitrarily large finite size; the overlap factor #(T∩(K−y)) can be unbounded, and an L1 function can place mass exactly on those overlaps so that the absolute series of integrals diverges even though the pointwise series converges. Corollary 3.3 and Proposition 3.2 then have no standing, and the conclusion supp(bf)⊂B(0,p^{nν+1}) does not follow. The final step also uses the converse of the stated Proposition 2.1 (bf compact support ⇒ f uniformly locally constant), which is not stated or proved. These are fixable if a dominated-convergence lemma is supplied, but as written the central argument has a load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":17283,"tokens_out":22368,"duration_ms":222202,"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":[{"comment":"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.","section":"Section 3, proof of Theorem 1.1"},{"comment":"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.","section":"Section 3.1, Proposition 3.1(2), Case n_nu = -infinity"},{"comment":"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.","section":"Sections 5.2 and 6.1, Theorems 5.3, 5.4 and 1.3"},{"comment":"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.","section":"Section 6.2, proof of Corollary 1.4"},{"comment":"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.","section":"Section 6.2, proof of Corollary 1.4"}],"minor_comments":[{"comment":"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).","section":"Theorem 4.1(2)"},{"comment":"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.","section":"Section 4.3, Eq. (4.7)"},{"comment":"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.","section":"Section 3.1, Eq. (3.3)"},{"comment":"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.","section":"Section 4.2, proof of Proposition 4.3"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the Fourier-distribution strategy is appropriate, but the proof currently has a genuine gap in passing to the Fourier side under only pointwise absolute convergence. This may be repaiable by adding an integral-absolute-convergence hypothesis or a substantial lemma, but the theorem as stated needs revision. A second concern is the heavy reliance on the author's own unpublished preprint [7] for the p=2 classification; I would ask the editor to require that those results be made available in refereed form or proved in the paper before a final decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has one genuinely valuable core: Theorem 1.1, extending Fan-Fan-Liao-Shi from indicator tilings to L1 functions with finitely many integer coefficients, and the zero-set description of the Fourier transform of a locally finite discrete measure. The applications to uniform partitions of unity and to tiles in Qp×Z/2Z are real and new, conditional on the core. The paper deserves engagement, but as written it is not rigorous enough to accept.\n\nThe main soft spot is exactly the one the stress-test flags. In Section 3 the proof moves from the pointwise a.e. equation sum_t v_t f(x−t)=w to the distribution identity f*ν=w and then to bf·bν=wδ0. That step needs a justification that f*ν is a well-defined Bruhat-Schwartz distribution. Pointwise absolute convergence of the tiling sum does not imply absolute convergence of sum_t v_t ∫ f(x)φ(x+t)dx; the overlap factor #(T∩(K−x)) is finite for each x but can be unbounded as x varies, and L1 mass can sit exactly on those overlaps. So Corollary 3.3 and the conclusion supp(bf)⊂B(0,p^{nν+1}) do not follow as written. This looks fixable—a truncation or dominated-convergence lemma should do—but it is load-bearing.\n\nSecond, the final step of Theorem 1.1 uses the statement that bf has compact support implies f is uniformly locally constant. That is true, but it is not what Proposition 2.1 says, and it is neither stated nor proved in the paper. Add it.\n\nThe remaining concerns are smaller. Theorem 1.3 is stated without a finite-measure assumption on Ω, and in the μT0=μT1 case the conclusion that Ω0∩Ω1 has measure zero needs the argument that a nonnegative integer-valued function tiling at level 1 cannot take value 2 on a positive measure set. The p=2 classification depends on the unpublished preprint [7], and the spectral construction in Corollary 1.4 is asserted rather than verified. All fixable, but they should be explicit.\n\nWho is this for? p-adic harmonic analysts and people working on Fuglede's conjecture. If the distribution gap is closed, Theorem 1.1 is a solid structural result. As it stands, I would not cite it. But it deserves a serious referee, not a desk reject; the referee should be asked to insist on the missing justifications.","headline":"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.","tokens_in":17863,"tokens_out":7480,"would_cite":false,"duration_ms":82263,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["43A70"],"pacs":[],"model":"deepseek-v4-flash","headline":"Any integrable function that tiles the p-adic field by translations is uniformly locally constant.","keywords":["periodic","translation tile","spectral set","p-adic field","uniformly locally constant","uniform partition of unity","Fuglede conjecture","p-homogeneous tree"],"falsifier":"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.","tokens_in":16812,"feed_emoji":"🧩","tokens_out":13733,"duration_ms":134029,"temperature":0.7,"pith_summary":"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.","feed_headline":"Tiling by a function on Q_p forces local constancy","feed_subtitle":"The result answers a partition-of-unity question on p-adic fields and ties tiles to spectra.","key_machinery":"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 $f$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\mathbb{Q}_p$."],"supporting_citations":[{"why":"Supplies the duality that compact support of $\\hat{f}$ is equivalent to $f$ being uniformly locally constant, and the prior result that Fuglede's conjecture holds in $\\mathbb{Q}_p$.","marker":"[5]"},{"why":"Provides the rotation-invariance lemma for zero sums of $p$-adic characters and the $p$-homogeneous characterization of spectral sets and tiling complements used in Theorem 1.3 and Corollary 1.4.","marker":"[6]"},{"why":"Gives the structural classification of tiles in $\\mathbb{Z}/p^n\\mathbb{Z} \\times \\mathbb{Z}/p\\mathbb{Z}$ that is lifted to $\\mathbb{Q}_p \\times \\mathbb{Z}/2\\mathbb{Z}$ in Theorem 1.3.","marker":"[7]"},{"why":"Schoenberg's cyclotomic polynomial theorem (Lemma 2.2) forces equal coefficients in vanishing sums of $p$-power roots of unity, underlying the $p$-cycle decomposition.","marker":"[26]"},{"why":"Supplies the $p$-adic distribution theory, including Proposition 2.9 identifying convolution with multiplication of Fourier transforms, used to pass $f*\\nu=w$ to $\\hat{f}\\cdot\\hat{\\nu}=w\\delta_0$.","marker":"[30]"},{"why":"Provides the theory of $p$-adic distributions used in the convolution and multiplication definitions, including Proposition 2.8 on convolution with test balls.","marker":"[1]"},{"why":"Leptin and Müller's paper poses the uniform-partition-of-unity question that Theorem 1.2 answers for $\\mathbb{Q}_p$ and supplies the line-group analogue.","marker":"[22]"}],"fun_headline_variants":["Tiling Q_p forces uniform local constancy","p-adic tiling functions are locally constant","Leptin-Muller question answered via tiling","Tiling p-adics ties to Fuglede conjecture","Tiles in Q_p x Z/2Z are spectral sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Tiling Q_p forces uniform local constancy","p-adic tiling functions are locally constant","Leptin-Muller question answered via tiling","Tiling p-adics ties to Fuglede conjecture","Tiles in Q_p x Z/2Z are spectral sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1492,"prompt_tokens":910,"completion_tokens":582,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":504}},"tokens_in":526,"tokens_out":582,"duration_ms":6194,"temperature":1.0,"reasoning_tokens":504,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:03:54.315120+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Fuglede’s conjecture holds in Qp","cited_arxiv_id":null,"evidence_quote":"Supplies the duality that compact support of $\\hat{f}$ is equivalent to $f$ being uniformly locally constant, and the prior result that Fuglede's conjecture holds in $\\mathbb{Q}_p$."},{"cited_title":"Compact open spectral sets in Qp","cited_arxiv_id":null,"evidence_quote":"Provides the rotation-invariance lemma for zero sums of $p$-adic characters and the $p$-homogeneous characterization of spectral sets and tiling complements used in Theorem 1.3 and Corollary 1.4."},{"cited_title":"The structure of tiles in $\\mathbb{Z}_{p^n}\\times \\mathbb{Z}_q$ and $\\mathbb{Z}_{p^n}\\times \\mathbb{Z}_p$","cited_arxiv_id":"2411.02696","evidence_quote":"Gives the structural classification of tiles in $\\mathbb{Z}/p^n\\mathbb{Z} \\times \\mathbb{Z}/p\\mathbb{Z}$ that is lifted to $\\mathbb{Q}_p \\times \\mathbb{Z}/2\\mathbb{Z}$ in Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Schoenberg's cyclotomic polynomial theorem (Lemma 2.2) forces equal coefficients in vanishing sums of $p$-power roots of unity, underlying the $p$-cycle decomposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the $p$-adic distribution theory, including Proposition 2.9 identifying convolution with multiplication of Fourier transforms, used to pass $f*\\nu=w$ to $\\hat{f}\\cdot\\hat{\\nu}=w\\delta_0$."},{"cited_title":"Albeverio, A","cited_arxiv_id":null,"evidence_quote":"Provides the theory of $p$-adic distributions used in the convolution and multiplication definitions, including Proposition 2.8 on convolution with test balls."},{"cited_title":"Uniform partitions of unity on locally compact groups","cited_arxiv_id":null,"evidence_quote":"Leptin and Müller's paper poses the uniform-partition-of-unity question that Theorem 1.2 answers for $\\mathbb{Q}_p$ and supplies the line-group analogue."}],"review_version":1}