{"id":"e5ea615a-2e13-458a-8183-c24d975b897b","arxiv_id":"2506.06670","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Infinite convolutions generated by equivalent digit sequences converge together, preserve equi-positivity, and under admissible-pair conditions admit a common spectrum, yielding non-compactly supported spectral measures in R^d.","lead":"Mathematicians prove conditions under which infinite convolutions of discrete measures in d-dimensional space exist and have an orthogonal Fourier basis, even when the support is unbounded. The result gives a new way to build non-compact spectral fractal measures and shows that equivalent digit sets produce measures sharing the same spectrum.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Explicit Λ in Theorem 2.6 is not justified: the proof uses containment L_{p,q}⊆R^T_{p,q}[-1/2,1/2)^d, which fails (e.g. R=4, L={0,2}); the core spectrality-in-Z^d claim remains plausible.","rationale":"The paper's main theorem, Theorem 2.6, aims to show that under RBC, uniform contractivity, and the existence of a PCC subsequence, the infinite convolution exists and is spectral with a spectrum in Z^d. The proof of equi-positivity is essentially correct: the tail product is bounded below uniformly on the box using PCC, the finite product is controlled by the contraction, and the transfer back to the original digit sets via total-variation closeness is legitimate after choosing K large. The reader's identified notational error about ξ_0 is best read as a typo, because the line defining ξ_j is correct. However, the 'Moreover' part of Theorem 2.6 asserts a concrete spectrum Λ built from the L_k sets. The proof of this part relies on the containment L_{m_{j−1},m_j} ⊆ R^T_{m_{j−1},m_j}[-1/2,1/2)^d, which is demonstrably false in a simple scalar example. Since this containment is used to conclude that all integer shifts k_{λ,j} vanish, the explicit formula for Λ is not established by the given argument. The main spectrality result in Z^d, which is what the reader identified as the strongest claim, still follows from the equi-positive family and Theorem 3.4; thus the verdict should remain conditional rather than unconditional accept or reject. The paper would be strengthened by either proving the explicit spectrum formula by a different route or deleting/qualifying the 'Moreover' clause.","tokens_in":17849,"tokens_out":51164,"duration_ms":442739,"concrete_test":"For d=1, set R_1=R_2=4 and L_1=L_2={0,2}. Compute L_{1,2}={0,2,8,10} and R_{1,2}^T[-1/2,1/2)=[-8,8); observe that the containment 10∈[-8,8) fails. Then, with B_k={0,1}, test whether the asserted Λ=∪_k{L_1+4L_2+...+4^{k-1}L_k} still satisfies Q_{μ,Λ}(ξ)≡1 for the infinite convolution μ. If Q≡1, the explicit formula survives but requires a new proof; if Q≠1, the 'Moreover' clause of Theorem 2.6 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central spectrality assertion of Theorem 2.6 is supported by a coherent equi-positivity argument; the index issue flagged by the reader (ξ_0 = R_{n_k}^{-T} ξ) is a typo, since the definition before it is correct. The load-bearing weakness lies in the 'Moreover' part of Theorem 2.6. The proof asserts L_{m_{j−1},m_j} ⊆ R^T_{m_{j−1},m_j}[-1/2,1/2)^d and uses this to set k_{λ,j}=0 for every λ, yielding the explicit spectrum Λ = ∪_k {L_1+R_1^T L_2+...+(R_{k-1}...R_1)^T L_k}. This containment is false in general. For d=1, take R=4 and L_i={0,2}; then L_{1,2}=L_1+4L_2={0,2,8,10}, while R^T_{1,2}[-1/2,1/2)=16[-1/2,1/2)=[-8,8), so 10∉[-8,8). Consequently, the assertion k_{λ,j}=0 for all λ∈L_{m_{j−1},m_j} is not justified; the spectrum constructed in Theorem 3.4 may contain additional integer translates, and the simple product formula for Λ is unproven. This does not invalidate the main conclusion that a spectrum in Z^d exists, but it means Theorem 2.6 as stated overclaims the explicit spectrum unless the proof is repaired or the 'Moreover' clause is weakened.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies infinite convolutions of uniform discrete measures in R^d, allowing non-compact support. It introduces an equivalence relation on sequences of finite digit sets, proves that equivalent sequences yield simultaneously convergent convolutions and simultaneously equi-positive families, and shows that equivalent admissible-pair sequences with congruent digit sets share a common spectrum. It gives a sufficient condition (remainder bounded condition plus uniform contractivity) for existence of such convolutions via a distributional three-series theorem, and a sufficient condition (existence of a subsequence satisfying a partial concentration condition) for spectrality. The main theorem asserts that under these hypotheses the infinite convolution is a spectral measure with a spectrum in Z^d, with an explicit product-form spectrum when each Hadamard triple has 0 in the dual digit set contained in the corresponding cube. An example of a non-compactly supported spectral measure is constructed.","tokens_in":18183,"tokens_out":15347,"duration_ms":124283,"significance":"If the main theorem is correct, it provides the first general construction of spectral measures arising as infinite convolutions of admissible pairs with non-compact support, extending the compact-support theory of Strichartz, Dutkay-Lai, An-Fu-Lai, and Dutkay-Haussermann-Lai. The equivalence and common-spectrum results are clean and potentially useful tools. The proofs are largely self-contained and do not rely on fitted parameters; Theorem 2.2 is a crisp Borel-Cantelli argument, Theorem 2.5 is a correct use of the Jessen-Wintner three-series criterion, and the equi-positivity transfer in Corollary 2.3 is sound. However, the explicit spectrum formula in the 'Moreover' part of Theorem 2.6 rests on a containment that is false in general, so the advertised explicit description of a spectrum is not currently established.","major_comments":[{"comment":"The proof asserts 'Since L_{m_{j-1},m_j} \\subseteq R^T_{m_{j-1},m_j}[-1/2,1/2)^d' and concludes that k_{\\lambda,j}=0 for all \\lambda\\in L_{m_{j-1},m_j}, which yields the simple product formula \\Lambda=\\cup_{k=1}^\\infty{L_1+R_1^T L_2+\\cdots+(R_{k-1}\\cdots R_1)^T L_k}. This containment does not follow from the hypothesis 0\\in L_k\\subseteq R_k^T[-1/2,1/2)^d for each k. For d=1, take R_1=R_2=4 and L_1=L_2={0,2}; then each L_k satisfies the hypothesis, but L_{1,2}=L_1+4L_2={0,2,8,10}, while R^T_{1,2}[-1/2,1/2)=16[-1/2,1/2)=[-8,8), so 10 is not contained. Consequently the step k_{\\lambda,j}=0 for all \\lambda\\in L_{m_{j-1},m_j} is unjustified, and the stated explicit spectrum \\Lambda is unproven. The main conclusion that a spectrum in Z^d exists does not depend on this containment, since Theorem 3.4 constructs a spectrum inductively; however, the 'Moreover' claim as stated overreaches. It should either be weakened to assert only the existence of a spectrum in Z^d (with the inductive construction from Theorem 3.4), or be supplemented by an additional hypothesis that ensures the containment.","section":"Section 6, proof of Theorem 2.6, 'Moreover' paragraph"},{"comment":"The example explicitly identifies \\Lambda = \\cup_{k=1}^\\infty{L_1+R_1^T L_2+\\cdots+(R_{k-1}\\cdots R_1)^T L_k} as a spectrum, relying on the unproven 'Moreover' clause of Theorem 2.6. Since the containment used to justify this formula is false in general and is not separately verified for the particular digit sets of the example, the claim that this specific \\Lambda is a spectrum is not established. The non-compactness argument and the existence of some spectrum in Z^d remain valid through the first part of Theorem 2.6, so the example can be repaired by removing the explicit \\Lambda, or by proving the containment for this specific construction. The paper should be revised accordingly.","section":"Example 2.7"}],"minor_comments":[{"comment":"The word 'spetral' should be 'spectral' in the sentence 'we show that the infinite convolution \\mu is a spetral measure'.","section":"Section 2, paragraph before Theorem 2.6"},{"comment":"The domain of x is written as '[0,1)^d' twice, whereas Definition 3.3 uses '[-1/2,1/2)^d'. The property is equivalent up to an integer shift, but the notation should be consistent with the definition.","section":"Proof of Lemma 3.5"},{"comment":"The text says 'It remains to estimate M_{B_{n_k}}(\\xi_0)' and later writes '|M_{B_{n_k}}(\\xi_0)| \\geq ...', but the factor in the product expansion is M_{B'_{n_k}}(\\xi_0), and the subsequent estimates use the sets B'^l_{n_k,1} and B'^l_{n_k,2}. This is a notational slip: the estimate should apply to M_{B'_{n_k}}(\\xi_0).","section":"Proof of Theorem 2.6, estimate after (6.4)"},{"comment":"The bound on the finite product is stated 'for all k,j>0', but the product is over j=1,\\ldots,J-1 and the bound depends on J; the range of j should be specified as 1\\le j\\le J-1.","section":"Proof of Theorem 2.6, inequality (6.4)"}],"recommendation":"major_revision","confidential_remarks":"The core spectrality result (existence of a spectrum in Z^d) is supported by a coherent equi-positivity argument and appears salvageable. The main weakness is localized to the explicit spectrum formula in the 'Moreover' clause of Theorem 2.6 and its use in Example 2.7; the authors should either prove the containment under additional assumptions or weaken the claim. I see no issue of circularity or hidden parameters. The fit with math.FA is appropriate, and the novelty with respect to non-compact spectral measures is real."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper is worth engaging. The main result—that a non-compactly supported infinite convolution in R^d is spectral if a subsequence satisfies the partial concentration condition—appears to go through. The genuinely new content is Definition 2.1 of equivalent digit sets, the common-spectrum theorem (Theorem 2.4), and the PCC condition that produces non-compact spectral measures. Theorem 2.5's existence proof via Kolmogorov's three-series theorem is clean, and the equi-positivity transfer in Corollary 2.3 is sound. Example 2.7 is a nice explicit non-compact spectral measure.\n\nNow the soft spot. The load-bearing flaw I see is in the 'Moreover' clause of Theorem 2.6. The proof needs the containment L_{m_{j−1},m_j} ⊆ R^T_{m_{j−1},m_j}[-1/2,1/2)^d to force k_{λ,j}=0 and get the simple product formula for Λ. That containment is false. In one dimension, take R=4 and L_1=L_2={0,2}. Then L_{1,2}=L_1+4L_2={0,2,8,10}, while R^T_{1,2}[-1/2,1/2)=16[-1/2,1/2)=[-8,8), and 10 is outside. So the proof's assertion that k_{λ,j}=0 for every λ is unjustified. This does not sink the main spectrality claim—Theorem 3.4 still delivers a spectrum in Z^d, just possibly with extra translates—but it means Theorem 2.6 as stated overclaims the explicit spectrum. The 'Moreover' clause needs a repair or a weakening.\n\nTwo smaller notes. The abstract says equivalent sequences yield the same spectrum, but Theorem 2.4 additionally requires B'_k ≡ B_k mod R_k Z^d; that condition is omitted in the abstract. Also, the ξ_0 indexing in the proof of Theorem 2.6 is fine as written; the definition ξ_0=R_{n_k}^{-T}ξ is consistent with the estimate (6.5).\n\nOverall, the paper shows honest, careful work and makes a real advance in the spectral theory of non-compact infinite convolutions. Send it to referees. It deserves serious review, but the explicit-spectrum formula needs fixing before the paper can be accepted; this is a major revision, not a rejection.","headline":"Main spectrality theorem is likely correct, but the explicit spectrum formula in Theorem 2.6 is not justified; worth refereeing with major revision.","tokens_in":18767,"tokens_out":7520,"would_cite":true,"duration_ms":58373,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A80","42C30","60B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that certain infinite convolutions in $\\mathbb{R}^d$, even with unbounded support, are spectral measures with spectra contained in $\\mathbb{Z}^d$.","keywords":["infinite convolutions","spectral measures","Hadamard triples","admissible pairs","equi-positive families","non-compact support","Fourier transform","fractal measures"],"falsifier":"The sharpest test would be a direct counterexample to Theorem 2.6: a sequence of admissible pairs satisfying the remainder bounded condition, uniform contraction, and the partial concentration condition on some subsequence, yet whose infinite convolution is not spectral.","tokens_in":17600,"feed_emoji":"📐","tokens_out":5094,"duration_ms":47140,"temperature":0.7,"pith_summary":"This paper studies infinite convolutions of discrete measures in $\\mathbb{R}^d$ that need not be compactly supported. It introduces a notion of equivalence for digit-set sequences based on finitely accumulating relative mismatch, and proves that equivalent sequences produce infinite convolutions that converge together and share equi-positivity and spectra. The main result gives sufficient conditions, namely remainder boundedness, uniform contraction, and a partial concentration condition on a subsequence, under which the infinite convolution exists and is a spectral measure with a spectrum in $\\mathbb{Z}^d$. The authors exhibit a two-dimensional example with unbounded support, showing the theorem reaches beyond the compactly supported fractal measures studied previously.","feed_headline":"Unbounded infinite convolutions can carry spectra","feed_subtitle":"A partial concentration condition plus bounded remainders guarantees an orthonormal exponential basis in Z^d.","key_machinery":"The argument rests on four objects: admissible pairs and Hadamard triples, where a digit set $B$ and expansive matrix $R$ admit a dual set $L$ making a unitary matrix and supplying a spectrum for the one-step measure; the equivalence relation on digit sequences defined by summability of relative mismatch counts; the remainder bounded condition and the partial concentration condition, which control how much digit mass lies far from the origin after rescaling; and equi-positive families, collections of tail measures whose Fourier transforms stay uniformly bounded away from zero near the integer lattice. Lemma 6.1, a lower bound for the average of exponentials with arguments in an interval, converts the concentration estimate into the uniform Fourier lower bound that equi-positivity requires.","core_discovery":"On the paper's own terms, the central discovery is Theorem 2.6: if $\\{(R_k,B_k)\\}$ is a sequence of admissible pairs satisfying the remainder bounded condition and the uniform contractive condition, and some subsequence satisfies the partial concentration condition with some $l\\in(0,1)$, then the infinite convolution $\\mu$ exists and is a spectral measure admitting a spectrum in $\\mathbb{Z}^d$. The proof works by replacing each digit set by a congruent set inside $R_k[-\\tfrac12,\\tfrac12)^d$, showing that the replacement preserves existence and equi-positivity, estimating the Fourier product from below using the partial concentration condition to obtain an equi-positive family, and then running the standard equi-positivity-to-spectrum construction. The paper also proves simultaneous convergence, preservation of equi-positivity, and a common spectrum for equivalent digit-set sequences. Example 2.7 gives a concrete non-compactly supported spectral measure in $\\mathbb{R}^2$.","pith_inferences":["One might try relaxing the partial concentration condition to a logarithmic or averaged concentration condition, since the proof's Borel-Cantelli step suggests a weaker tail condition may still yield equi-positivity in some examples.","The equivalence relation on digit sets is metric-like and may connect to Wasserstein or Prokhorov stability of infinite convolutions, allowing perturbation results for spectra of random convolutions.","The construction suggests a route to non-compact spectral measures with prescribed dimension by choosing slowly growing digit sets that satisfy the partial concentration condition while pushing mass to infinity.","A direct consequence not stated in the paper is that any two equivalent admissible sequences sharing the same tail asymptotics will share an explicit common spectrum, not merely spectrality."],"forward_implications":["If two sequences of digit sets differ only by finitely accumulating relative mismatch, their infinite convolutions exist together and, under admissible-pair hypotheses, share a spectrum, so compact and non-compact examples can be analyzed simultaneously.","Theorem 2.5 supplies a new existence criterion for non-compactly supported infinite convolutions in any dimension.","Theorem 2.6 produces spectral measures with unbounded support, a class not covered by earlier compact-support theories.","The explicit Example 2.7 gives a concrete non-compact spectral measure in $\\mathbb{R}^2$ and shows the hypotheses are satisfiable.","Equivalent admissible sequences with congruent digits modulo $R_k\\mathbb{Z}^d$ have the same spectrum, so spectral data are stable under digit perturbations."],"supporting_citations":[{"why":"Supplies the Hadamard-triple lemmas used to construct spectra and to transfer spectrality under congruent digit sets.","marker":"[12]"},{"why":"Establishes the spectrality framework for compactly supported infinite convolutions generated by arbitrary and random convolutions.","marker":"[13]"},{"why":"Introduces the equi-positive family method that the paper adapts to non-compact infinite convolutions.","marker":"[1]"},{"why":"Provides the basic criterion that a set is a spectrum exactly when the associated exponential sum equals one pointwise.","marker":"[24]"},{"why":"Supplies the distribution version of Kolmogorov's three-series theorem used to prove existence of the infinite convolution.","marker":"[20]"},{"why":"Provides prior weak convergence and spectrality results for infinite convolutions that motivate the existence arguments.","marker":"[31]"},{"why":"Gives the one-dimensional admissible-pair spectrality result and the spectrum construction that the paper extends.","marker":"[29]"},{"why":"First studied spectrality of infinite convolutions, providing the historical and technical setting for the paper.","marker":"[39]"},{"why":"Supplies the one-dimensional partial concentration conditions that the paper generalizes to higher dimensions.","marker":"[33]"}],"fun_headline_variants":["Partial concentration yields spectra for noncompact convolutions","Admissible pairs and concentration give spectral measures in R^d","Unbounded infinite convolutions can be spectral via admissible pairs","Equi-positivity and concentration ensure spectra of infinite convolutions","New conditions for existence and spectrality of infinite convolutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that some subsequence of the digit sets satisfies the partial concentration condition: after rescaling by the inverse matrices, most digits lie close to the origin with spread strictly less than $1-l$, and the unscattered digits have summable relative counts.","fun_headline_variants_meta":{"raw":{"variants":["Partial concentration yields spectra for noncompact convolutions","Admissible pairs and concentration give spectral measures in R^d","Unbounded infinite convolutions can be spectral via admissible pairs","Equi-positivity and concentration ensure spectra of infinite convolutions","New conditions for existence and spectrality of infinite convolutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1281,"prompt_tokens":854,"completion_tokens":427,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":344}},"tokens_in":470,"tokens_out":427,"duration_ms":4756,"temperature":1.0,"reasoning_tokens":344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:55:05.875715+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The sharpest test would be a direct counterexample to Theorem 2.6: a sequence of admissible pairs satisfying the remainder bounded condition, uniform contraction, and the partial concentration condition on some subsequence, yet whose infinite convolution is not spectral.","supporting_citations":[{"cited_title":"Dutkay, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Hadamard-triple lemmas used to construct spectra and to transfer spectrality under congruent digit sets."},{"cited_title":"Dutkay, C.-K","cited_arxiv_id":null,"evidence_quote":"Establishes the spectrality framework for compactly supported infinite convolutions generated by arbitrary and random convolutions."},{"cited_title":"An, X.-Y","cited_arxiv_id":null,"evidence_quote":"Introduces the equi-positive family method that the paper adapts to non-compact infinite convolutions."},{"cited_title":"Jorgensen, S","cited_arxiv_id":null,"evidence_quote":"Provides the basic criterion that a set is a spectrum exactly when the associated exponential sum equals one pointwise."},{"cited_title":"Jessen, A","cited_arxiv_id":null,"evidence_quote":"Supplies the distribution version of Kolmogorov's three-series theorem used to prove existence of the infinite convolution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides prior weak convergence and spectrality results for infinite convolutions that motivate the existence arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the one-dimensional admissible-pair spectrality result and the spectrum construction that the paper extends."},{"cited_title":"Strichartz, Mock Fourier series and transforms associated with certain Cantor measures,J","cited_arxiv_id":null,"evidence_quote":"First studied spectrality of infinite convolutions, providing the historical and technical setting for the paper."},{"cited_title":"Existence and spectrality of infinite convolutions generated by infinitely many admissible pairs","cited_arxiv_id":"2312.16863","evidence_quote":"Supplies the one-dimensional partial concentration conditions that the paper generalizes to higher dimensions."}],"review_version":1}