{"id":"5b5e4a75-594d-4029-a17d-9715381e869c","arxiv_id":"1908.11147","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Niederreiter and Halton sequences fail to satisfy Poissonian pair correlations in any dimension, confirming Larcher and Stockinger's conjecture.","lead":"This paper proves that two standard multi-dimensional low-discrepancy sequences, Niederreiter and Halton sequences, do not have Poissonian pair correlations despite being uniformly distributed. It settles an open conjecture and gives a general criterion for detecting such non-random pair structure.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The column-by-column half of Lemma 2 is under-proved; if its L_f ≤ d f bound fails, the digital proof's difference-vector argument collapses.","rationale":"I read the paper in good faith and found the general strategy sound: Proposition 1 is a clean sufficient criterion, and its proof is correct. The Halton argument is detailed and the constants check out once the accumulation-point lemma is accepted; the one-line derivation of the corner accumulation from Minkowski is terse but standard. For the digital sequences, the Niederreiter part of Lemma 2 is cited, while the column-by-column part is asserted in a sentence that does not actually prove the row-length bound L_f ≤ d f. Since Theorem 1 depends on that bound to identify Δ, this is the weakest load-bearing step. My own partial re-derivation suggests the bound is true, and the rest of the digital proof (zero pattern, annulus constants, choice of epsilon) is consistent. Therefore I do not believe the theorem is false, but the manuscript should supply the missing derivation or an explicit reference for the column-by-column case; hence conditional acceptance rather than unconditional acceptance.","tokens_in":13936,"tokens_out":51913,"duration_ms":460910,"concrete_test":"Independently re-derive Lemma 2 for the column-by-column construction. For q=2, d=2, q1=x, q2=x^2+x+1 (e=(1,2), v=2), build C^(1), C^(2) and S from the monic p_k, and for f=2,4,6 check that the first f rows of C^(j)S have last non-zero column index < d f in all cases. Also verify the shape (0^{uvθ}, 1, 0^{uv−1}, ...) of C^(j)Δ for a representative n. In parallel, prove the general bound by showing that a nonzero entry in row i≤f at column k forces the q_j-adic exponent t_j to satisfy t_j v ≤ f−e_j, which yields k < f+v ≤ d f; if either check fails, the digital annulus count is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 2 is the linchpin of Theorem 1. It asserts that after right-multiplication by S the column-by-column generating matrices still form a digital (0,e,d)-sequence and satisfy the row-length bound L_f ≤ d f for v|f. The proof for the column-by-column construction is one sentence ('linearity ... and ... basis'), with no derivation of the L_f bound. In the proof of Theorem 1, this bound is exactly what makes the last column of D_{m×(m+1)} S_{m+1} zero, forcing S^{-1} Δ = (0,...,0, φ^{-1}(1))^T and hence Δ = φ^{-1}(1) times the p_m column. If the bound failed for some f, the explicit shape of C^(j)Δ used to produce the distance interval would fail, and the annulus count in Proposition 1 could not be established. I checked the bound heuristically via q_j-adic valuations (for a row i≤f, nonzeroness forces the q_j-adic exponents t_j ≤ f/v − 1, which yields last column index < df), and I found no counterexample; but this derivation is absent from the paper, so the column-by-column half of Lemma 2 is a genuine gap in exposition and the most load-bearing unverified step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves that two classical multidimensional low-discrepancy sequences—digital (0,e,d)-sequences generated by the Niederreiter construction or the alternative column-by-column construction, and Halton sequences in pairwise coprime integer bases—do not have Poissonian pair correlations under the d-dimensional sup-norm definition (1). The authors introduce a general criterion (Proposition 1): if along a subsequence N_k there are at least c N_k ordered pairs with sup-norm distance in an annulus (a/N_k^{1/d}, b/N_k^{1/d}] with c > (2b)^d - (2a)^d, then the sequence cannot have Poissonian pair correlations. They then verify this condition for both families. For digital sequences, they use a NUT-scrambling matrix S (Lemma 2) to force the difference vector between paired points to equal a fixed column of S, yielding a precise interval for the sup-norm distance. For Halton sequences, they construct many pairs n, n+M whose distance is controlled by the first coordinate, using congruences and an approximation lemma for irrational rotations (Lemma 3).","tokens_in":14165,"tokens_out":40673,"duration_ms":365893,"significance":"If correct, the paper closes a natural gap in the theory of pair correlations, confirming a conjecture of Larcher and Stockinger for Halton sequences and extending known one-dimensional negative results to two prominent multidimensional families. The general criterion in Proposition 1 is clean, rigorous, and likely to be reused. The Halton proof is self-contained, checks all constants explicitly, and verifies the required inequality (12). The digital proof is conceptually convincing and, modulo Lemma 2, provides the needed annulus count with explicit constants. The main weakness is that Lemma 2's column-by-column half is only sketched; because that lemma is load-bearing for Theorem 1, the manuscript as written is not fully complete, though the gap appears repairable.","major_comments":[{"comment":"For the column-by-column construction, the proof consists of a single sentence invoking 'linearity' and the fact that (1,x,x^2,...) and (1,p_1(x),p_2(x),...) are both bases. The crucial bound L_f ≤ d f for v|f is not derived. This bound is exactly what makes the last column of D_{m×(m+1)}S_{m+1} vanish in the proof of Theorem 1, forcing S^{-1}Δ = (0,...,0,φ^{-1}(1))^T and hence the explicit form of Δ used to derive the distance interval. If this bound were to fail, the annulus count in Proposition 1 could not be established. Please provide a complete derivation of the row-length bound for the column-by-column construction, or give a precise independent reference.","section":"Section 2.1, Lemma 2"}],"minor_comments":[{"comment":"The definition of L_f is somewhat ambiguous; please state explicitly that L_f is the maximum, over all first f rows of all generating matrices, of one plus the index of the last nonzero column, and specify how identically zero rows are treated.","section":"Section 2.1, after Eq. (3)"},{"comment":"In the paragraph after Eq. (6), the line 'if in (n)_{b1} we have that n_{uτ k 1} ≠ b_1 − 1' contains a typo; it should read n_{uτ_1k_1} ≠ b_1 − 1.","section":"Section 2.2, proof of Theorem 2"},{"comment":"The sentence 'if ε_j are chosen small enough the integers z_i^{(0)} will be distinct' is terse; the choice of N and ε_j must satisfy both the Minkowski volume condition and ε_j < 1/2, and distinctness follows because for a fixed z_0 the inequalities |α_j z_0 − z_j| ≤ ε_j determine z_j uniquely when ε_j < 1/2.","section":"Section 2.2, Lemma 3"},{"comment":"After selecting n with the maximal digit difference, the count leading to c = 1/q counts ordered pairs; it may help the reader to state explicitly that each selected pair (n,l) also contributes the reversed ordered pair (l,n).","section":"Section 2.1, proof of Theorem 1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: this is a significant paper that is likely correct, and I expect to be able to accept it after a revision. The only substantive issue is the under-proved Lemma 2 for the column-by-column construction; it is load-bearing for Theorem 1. The self-citation to [10] for the Niederreiter half is appropriate, and the column-by-column half should either be proved in detail or referenced to a fully verifiable source. The Halton part is solid and self-contained. The paper fits the journal's scope well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this paper gives a genuinely new negative result—Halton and Niederreiter sequences in higher dimensions do not have Poissonian pair correlations—and it does so with a general criterion that is clean enough to reuse. The results are not a repackaging of earlier 1D work. The criterion, Proposition 1, is quite elegant: it says enough pairs in a thin annulus around a fixed radius contradicts Poissonian scaling, with the constants chosen so the annulus count exceeds the difference between the two balls. That part is rigorous and complete.\n\nThe Halton proof is the stronger half. It spells out the construction of subintervals, the use of Minkowski to find accumulating fractional parts of logs, and the counting of good n; the constants a,b,c are explicit and condition (12) is verified. I checked the reasoning and it holds.\n\nThe digital-sequence proof is credible but noticeably terser. The main load-bearing point is Lemma 2, which says that after multiplying by the NUT matrix S, the column-by-column generating matrices still give a (0,e,d)-sequence with row-length bound L_f ≤ d f. For the Niederreiter construction this is cited from [10]; for the column-by-column construction it is justified in one sentence via linearity and basis change. On reading the actual proof of Theorem 1, yes, this bound is exactly what forces the last column of D S to be zero, so the entire shape of Δ depends on it. The stress-test note checked the valuation arithmetic and found no counterexample; I believe the bound is true, but the paper does not show it. That is a real gap in exposition, not a fatal error. A referee should ask for the derivation.\n\nThe other small thing: the 'at least Mm/q values of n' claim in Theorem 1 is asserted without proof, but that is a minor counting fact.\n\nCitation pattern is fine. The reliance on [10] is legitimate for the Niederreiter case. No self-citation inflation.\n\nWho should read this? People working on pair correlations of low-discrepancy sequences and uniform distribution—and anyone using Halton or Niederreiter in QMC who wants to know their theoretical fine-scale behavior. It is not a practical integration paper.\n\nMy call: send it to peer review. It deserves a serious referee. The verdict will likely be accept after the authors expand Lemma 2's proof and the one-sentence justifications around it.","headline":"A credible negative result that closes an open problem; the Halton proof is solid, and the digital proof has a terse but likely true step a referee should ask to be expanded.","tokens_in":14694,"tokens_out":2378,"would_cite":true,"duration_ms":23878,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11K38","11K31"],"pacs":[],"model":"deepseek-v4-flash","headline":"Niederreiter digital sequences and Halton sequences, though uniformly distributed, fail Poissonian pair correlations in every dimension.","keywords":["Poissonian pair correlations","Niederreiter sequences","Halton sequences","digital (0,e,d)-sequences","low-discrepancy sequences","uniform distribution","quasi-Monte Carlo","pair correlations"],"falsifier":"For a small explicit instance of the column-by-column construction (for example, $d=2$ over $\\mathbb{F}_2$ with $q_1(x)=x$ and $q_2(x)=x+1$), compute the transformed matrices $C^{(j)}S$ from Lemma 2 and test the row-length bound $L_f \\le d f$ for every $f$ that is a multiple of $v=\\operatorname{lcm}(e_1,e_2)$; if any such $f$ violates the bound, the sparse difference vector used in the proof of Theorem 1 is not available. A direct count of pairs in the annulus $(a/N^{1/d}, b/N^{1/d}]$ for $N=2q^m$ on the same instance should also reproduce the claimed excess of at least $cN$ pairs.","tokens_in":13722,"feed_emoji":"🎲","tokens_out":12893,"duration_ms":118971,"temperature":0.7,"pith_summary":"The paper proves that two standard families of uniformly distributed multi-dimensional point sequences—Niederreiter digital sequences and Halton sequences—fail the stronger property of Poissonian pair correlations. In the d-dimensional sup-norm sense, Poissonian pair correlations would require the normalized count of pairs closer than $s/N^{1/d}$ to approach $(2s)^d$, as it does for independent random points. Instead, along infinitely many block lengths, an excess of pairs lie in a narrow annulus of distances, and the paper's general criterion (Proposition 1) shows this annulus excess is incompatible with the Poissonian limit. This confirms a conjecture posed in earlier work and extends known one-dimensional negative results. The main tool is a self-contained criterion that turns a local regularity of a sequence into a proof of non-Poissonian behavior.","feed_headline":"Halton and Niederreiter sequences fail the random-pair test","feed_subtitle":"The paper proves these evenly spread point sets lack the random-like local pair statistics.","key_machinery":"The load-bearing object is Proposition 1, the annulus criterion: a sequence that, for infinitely many $N$, has at least $cN$ pairs at sup-norm distances in $(a/N^{1/d}, b/N^{1/d}]$ with $c>(2b)^d-(2a)^d$ cannot have Poissonian pair correlations. For digital sequences, the supporting machinery is a change-of-basis matrix $S$ (Lemma 2) that makes the difference vector between paired points explicit and sparse, so that the annulus count can be computed; this uses invariance of digital $(t,e,d)$-sequences under right multiplication by non-singular upper triangular (NUT) matrices. For Halton sequences, the supporting machinery is a Diophantine approximation lemma (Lemma 3) that finds simultaneous approximants of the numbers $\\log_{\\beta_j}(\\beta_1)$ accumulating at cube-corner values, which selects block lengths where the first coordinate dominates all others in the distance calculation.","core_discovery":"The central claim, stated as Theorems 1 and 2, is that digital $(0,e,d)$-sequences generated by the Niederreiter construction or by the alternative column-by-column construction, and Halton sequences in pairwise coprime integer bases, do not have Poissonian pair correlations under the sup-norm definition (1). These sequences are uniformly distributed, so the theorems separate uniform distribution from the stronger random-like local statistics measured by pair correlations. For each family the proof produces infinitely many block lengths $N$ for which at least $cN$ pairs have sup-norm distance in $(a/N^{1/d}, b/N^{1/d}]$ with $c > (2b)^d-(2a)^d$; by Proposition 1, such an excess of near-duplicate distances rules out Poissonian pair correlations.","pith_inferences":["Read as a sufficient condition, Proposition 1 offers a concrete test for non-Poissonian behavior: look for long blocks where many shifted pairs land in a narrow distance annulus, and other digit-based sequences beyond those treated here can be checked for the same signature.","The Halton proof's reliance on accumulation points at cube corners suggests that any family built from radical-inverse functions in coprime bases will exhibit structurally similar pair collisions, a prediction that could be tested on generalized Halton variants.","For quasi-Monte Carlo practice, the theorems imply that low discrepancy does not certify random-like local spacing; users who need Poissonian local statistics must look beyond these standard constructions."],"forward_implications":["Uniform distribution is strictly weaker than Poissonian pair correlations for these sequences: each family is uniformly distributed, yet fails the pair-correlation limit along an infinite subsequence of block lengths.","Digital $(0,e,d)$-sequences from the Niederreiter and column-by-column constructions, and Halton sequences in pairwise coprime bases, cannot serve as examples of d-dimensional sequences with Poissonian pair correlations.","The general criterion (Proposition 1) gives a template for analyzing additional $(t,s)$-sequence classes, such as generalized Niederreiter or Niederreiter-Xing constructions, by verifying the same annulus excess.","For Halton sequences, the proof is carried out without resolving an open problem on linear independence of numbers like $1/\\log 2$, $1/\\log 3$ and $1/\\log 5$; if that independence were known, the simultaneous-approximation step would simplify considerably."],"supporting_citations":[{"why":"defines the d-dimensional sup-norm Poissonian pair-correlation concept and provides the background on uniform distribution for that notion.","marker":"[7]"},{"why":"supplies the cited scrambling result that lets the transformed Niederreiter generating matrices stay within the digital (0,e,d)-sequence framework in Lemma 2.","marker":"[10]"},{"why":"provides the convex-body lattice-point theorem undergirding Lemma 3, which produces the simultaneous approximants used in the Halton proof.","marker":"[14]"},{"why":"introduces Halton sequences via radical-inverse functions in pairwise coprime bases, the family analyzed in Theorem 2.","marker":"[6]"},{"why":"gives the alternative column-by-column construction of generating matrices that Theorem 1 covers alongside the Niederreiter construction.","marker":"[8]"},{"why":"introduces the Niederreiter construction of generating matrices, the first family treated in Theorem 1.","marker":"[15]"},{"why":"contains the one-dimensional negative results that the paper extends to higher dimensions and invokes for the d=1 base case.","marker":"[12]"},{"why":"supplies the lemma on right-multiplication by non-singular upper triangular matrices preserving digital (t,e,d)-sequence properties, used in Lemma 2.","marker":"[4]"}],"fun_headline_variants":["Uniform but not Poissonian: Halton and Niederreiter fail pair test","Niederreiter and Halton: uniform but not Poissonian","Pair correlations prove Halton and Niederreiter non-Poissonian","Halton and Niederreiter: pair statistics not Poissonian","Uniform, but no Poissonian pairs for Halton and Niederreiter"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the technical lemma that a particular change of basis applied to the generating matrices preserves the digital $(0,e,d)$-sequence property and a row-length bound $L_f \\le d f$; for the Niederreiter construction the lemma is cited from earlier work, while for the column-by-column construction it is asserted in one sentence, and if it failed the explicit distance formulas used to count pairs would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Uniform but not Poissonian: Halton and Niederreiter fail pair test","Niederreiter and Halton: uniform but not Poissonian","Pair correlations prove Halton and Niederreiter non-Poissonian","Halton and Niederreiter: pair statistics not Poissonian","Uniform, but no Poissonian pairs for Halton and Niederreiter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001493,"raw_usage":{"total_tokens":5913,"prompt_tokens":786,"completion_tokens":5127,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":402,"completion_tokens_details":{"reasoning_tokens":5033}},"tokens_in":402,"tokens_out":5127,"duration_ms":32720,"temperature":1.0,"reasoning_tokens":5033,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:23:25.311775+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small explicit instance of the column-by-column construction (for example, $d=2$ over $\\mathbb{F}_2$ with $q_1(x)=x$ and $q_2(x)=x+1$), compute the transformed matrices $C^{(j)}S$ from Lemma 2 and test the row-length bound $L_f \\le d f$ for every $f$ that is a multiple of $v=\\operatorname{lcm}(e_1,e_2)$; if any such $f$ violates the bound, the sparse difference vector used in the proof of Theorem 1 is not available. A direct count of pairs in the annulus $(a/N^{1/d}, b/N^{1/d}]$ for $N=2q^m$ on the same instance should also reproduce the claimed excess of at least $cN$ pairs.","supporting_citations":[{"cited_title":"Hinrichs, L","cited_arxiv_id":null,"evidence_quote":"defines the d-dimensional sup-norm Poissonian pair-correlation concept and provides the background on uniform distribution for that notion."},{"cited_title":"Hofer and G","cited_arxiv_id":null,"evidence_quote":"supplies the cited scrambling result that lets the transformed Niederreiter generating matrices stay within the digital (0,e,d)-sequence framework in Lemma 2."},{"cited_title":"Minkowski, Geometrie der Zahlen , Chelsea Publishing Company, reprint 1953","cited_arxiv_id":null,"evidence_quote":"provides the convex-body lattice-point theorem undergirding Lemma 3, which produces the simultaneous approximants used in the Halton proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces Halton sequences via radical-inverse functions in pairwise coprime bases, the family analyzed in Theorem 2."},{"cited_title":"Hofer, A construction of low-discrepancy sequences i nvolving finite-row digital (t, s )-sequences, Monatshefte für Mathematik , 171(1):77-89, 2013","cited_arxiv_id":null,"evidence_quote":"gives the alternative column-by-column construction of generating matrices that Theorem 1 covers alongside the Niederreiter construction."},{"cited_title":"Niederreiter, Low-discrepancy and low-dispersion sequences, Journal of Num- ber Theory, 30(1):51-70, 1988","cited_arxiv_id":null,"evidence_quote":"introduces the Niederreiter construction of generating matrices, the first family treated in Theorem 1."},{"cited_title":"Some negative results related to Poissonian pair correlation problems","cited_arxiv_id":"1803.05236","evidence_quote":"contains the one-dimensional negative results that the paper extends to higher dimensions and invokes for the d=1 base case."},{"cited_title":"Faure and S","cited_arxiv_id":null,"evidence_quote":"supplies the lemma on right-multiplication by non-singular upper triangular matrices preserving digital (t,e,d)-sequence properties, used in Lemma 2."}],"review_version":1}