{"id":"343cefa8-3d84-428c-8480-a3ba06e082fb","arxiv_id":"2506.01736","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Rough-number rotations by badly approximable alpha have Poissonian correlations of all orders and Poissonian gaps, while almost every alpha makes the correlations non-Poissonian.","lead":"This paper proves that a sequence built from rough integers, when multiplied by any badly approximable irrational number and reduced modulo one, has Poissonian correlations of every order, the first explicit example of this kind. It also shows the property fails for almost every irrational, disproving a 2020 conjecture.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5.3's regularity assertion is false as stated: r(x) may drop sharply, so log f(x/(log x)^{k+1})/log f(x) does not tend to 1 and the sequence-to-array reduction fails.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: the transition from uniform triangular arrays to the actual roughness sequence in Section 5.3 relies on the assertion log f(x/(log x)^{k+1})/log f(x) → 1, which does not follow from the stated monotonicity and lower-bound hypotheses. My analysis confirms that the assertion is not merely unproved but false: monotonically decreasing r may include arbitrarily rapid downward jumps while still satisfying r(x) ≫_A (log log x)^A/log x for every A. The explicit step-function example above yields infinitely many x where the ratio is unbounded, so Φ(x, f(x)) and Φ(x, f(x/(log x)^{k+1})) are not asymptotically equal. Since Section 5.3 uses this equality to sandwich the sequence-level correlations between two triangular-array limits with matching normalizations, the final limit could differ from Vol(R); the proof of Theorem 1 as stated is therefore incomplete in its central reduction. The rest of the paper—especially the Bohr-set equidistribution theorem, the sieve estimate, and the averaging lemma—appears plausible and may support a corrected theorem under an additional regularity hypothesis on r, but as written the main result is not established. No change to the reader's REJECT verdict is warranted.","tokens_in":30398,"tokens_out":8915,"duration_ms":96219,"concrete_test":"Construct r as in the attack: fix g(x) = exp(−(log log x)^2), choose x_n with x_n/(log x_n)^{k+1} > x_{n−1} and g(x_{n−1})/g(x_n) = n, and set r(x) = g(x_n) on [x_n, x_{n+1}). At x = x_n, compute log f(x_n/(log x_n)^{k+1})/log f(x_n) and show it is n(1+o(1)), so it diverges. Then use Mertens' theorem to estimate Φ(x_n, f(x_n/(log x_n)^{k+1}))/Φ(x_n, f(x_n)) and verify it is asymptotic to log f(x_n/(log x_n)^{k+1})/log f(x_n), hence unbounded, contradicting the Section 5.3 claim. This directly settles whether the stated hypotheses suffice for the sequence-to-array reduction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reduction in Section 5.3 requires Φ(x, f(x)) ∼ Φ(x, f(x/(log x)^{k+1})), justified by 'since log z / log z⁻ tends to 1'. But this ratio need not tend to 1 under the stated hypotheses. Let g(x) = exp(−(log log x)^2), which satisfies g(x) ≫_A (log log x)^A/log x for every A. Choose x_n increasing so fast that x_n/(log x_n)^{k+1} > x_{n−1} and g(x_{n−1})/g(x_n) = n. Define r(x) = g(x_n) for x ∈ [x_n, x_{n+1}). Then r decreases monotonically to 0 and obeys the lower bound, yet at x = x_n we have r(x_n) = g(x_n) while r(x_n/(log x_n)^{k+1}) = g(x_{n−1}), so log f(x_n/(log x_n)^{k+1})/log f(x_n) = n(1+o(1)) → ∞, not tending to 1. Consequently Φ(x, f(x))/Φ(x, f(x/(log x)^{k+1})) ∼ log f(x/(log x)^{k+1})/log f(x) is unbounded along this sequence. The squeeze in Section 5.3 therefore compares two triangular arrays with different normalizations, and the claimed Poissonian limit for the actual sequence does not follow from the proved triangular-array result. A regularity condition such as r(x)/r(x/(log x)^{k+1}) → 1 is needed; without it Theorem 1 is not established as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies fine-scale statistics of the sequence (a_n^{(f)} α) mod 1, where a_n^{(f)} is the n-th integer whose least prime factor exceeds f(n). For f(x)=x^{r(x)} with r decreasing to 0 and r(x) ≫_A (log log x)^A / log x for every A>0, Theorem 1 claims that for every badly approximable α the sequence has Poissonian correlations of all orders, hence Poissonian gaps; the same is claimed for the associated triangular arrays. Theorem 2 asserts a converse for α satisfying liminf n ∥nα∥ log f(n)=0, and in particular that the result fails for Lebesgue-almost every α, disproving a conjecture of Larcher and Stockinger. The proof combines a sieve estimate (Lemma 4), an equidistribution theorem for Bohr sets modulo d (Theorem 3), proved via Ostrowski expansions and Markov chains, a key averaging lemma (Lemma 16), and a final reduction from triangular arrays to the actual sequence in Section 5.3.","tokens_in":30642,"tokens_out":10903,"duration_ms":109728,"significance":"If the main theorem were established, it would be a notable advance: it would give the first explicit sequence of the form {a_n α} with Poissonian correlations of all orders, confirm a rough-number analogue of the Rudnick–Sarnak heuristic, and settle the Larcher–Stockinger conjecture in the negative. The paper also contains a substantial and partly novel technical apparatus: the Bohr-set divisibility theorem (Theorem 3), the sieve estimate Lemma 4, and the averaging machinery of Section 5.1 are developed in considerable detail and are plausibly correct. The Markov-chain approach to equidistribution in Bohr sets is a genuine methodological contribution that may be of independent interest. However, the advertised sequence-level result is not proven as stated because the passage from triangular arrays to the sequence in Section 5.3 contains a load-bearing unproved assertion.","major_comments":[{"comment":"The reduction from triangular arrays to the actual sequence is invalid under the stated hypotheses. The text asserts: 'Observe that Φ(x,z)≤N(x,f)≤Φ(x,z−) and Φ(x,z)∼Φ(x,z−) since log z / log z− → 1.' The implication 'log z / log z− → 1' does not follow, and is in fact false for admissible r. Let L=(log x)^{k+1}. The lower-bound hypothesis r(x) ≫_A (log log x)^A / log x for all A does not prevent r from dropping sharply. For example, set g(t)=exp(−√(log log t)) and choose x_n so that x_n/L_n > x_{n−1} and g(x_{n−1})/g(x_n)=n, where L_n=(log x_n)^{k+1}; define r(x)=g(x_n) on [x_n,x_{n+1}). This r decreases monotonically to 0 and satisfies the lower bound, but at x=x_n we have log f(x_n/L_n)/log f(x_n)=n(1+o(1))→∞. Hence Φ(x,f(x)) and Φ(x,f(x/L)) need not be asymptotically equal, and the two triangular-array limits in the squeeze are normalized by different, incompatible quantities. The proof of Theorem 1 therefore does not establish the claimed sequence result under the stated assumptions. A regularity condition such as r(x)/r(x/(log x)^{k+1})→1 appears to be necessary, and must be added to the theorem or proved from a strengthened hypothesis.","section":"Section 5.3"},{"comment":"Theorem 2 is first proved for triangular arrays, and the transition to the actual sequence is explicitly delegated to 'the step to move to the sequence can be established as in Section 5.3.' Since the Section 5.3 reduction is not valid under the stated assumptions on r, the sequence version of Theorem 2 is not established either. The statement of Theorem 2, as well as the assertion that it disproves the Larcher–Stockinger conjecture for the sequence (a_n^{(f)} α), is therefore currently unsupported.","section":"Section 6"}],"minor_comments":[{"comment":"The definition of Φ(x,z) reads '#{1≤n≤x : P^−(x)>z}'; the argument of P^− should be n, not x.","section":"Definition 1"},{"comment":"The displayed statement '#B(x,I_x)∼λ(I_x) I_x' should be '#B(x,I_x)∼λ(I_x) x'; the proof and the surrounding text use λ(I_x)x.","section":"Lemma 11"},{"comment":"The sentence 'We denote by {x} the integer part of x' should read 'the fractional part of x'; the standard notation {x} is used later with that meaning.","section":"Notation, Section 2.0.1"},{"comment":"The sentence 'since log z / log z− → 1' should also specify that z and z− depend on x and that the limit is taken as x→∞; as written it is ambiguous.","section":"Section 5.3"},{"comment":"The proof of Proposition 13 labels the second inequality as '(ii)' and then refers to '(iii)' for the digit form; the labels should be rechecked for consistency.","section":"Proposition 13"}],"recommendation":"major_revision","confidential_remarks":"The central issue is isolated to Section 5.3, but it is load-bearing: the main theorem's sequence-level statement is not proved without an additional regularity assumption on r. If the author can prove the needed ratio condition under a reasonable strengthening of the hypotheses, the paper would still be a strong contribution. I also note that Proposition 12 is quoted from the author's companion preprint [20]; if that preprint is not yet accepted, the proof of Theorem 3 is conditional on an external unpublished result, which should be clarified. I do not see circularity or fabricated entities in the derivation, but the advertised result is currently overclaimed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is not a crank paper. It is the first explicit sequence of the form {a_n alpha} with Poissonian correlations of all orders for badly approximable alpha, plus a disproof of the Larcher--Stockinger conjecture. The triangular-array part looks well-built: the sieve lemma, the Bohr-set equidistribution theorem (Theorem 3), and the averaging lemma (Lemma 16) are developed carefully, and the Markov-chain argument is original and plausible. The paper is also honest about what it borrows, e.g., Proposition 12 from the author's companion preprint.\n\nThe problem is the last step. Section 5.3 needs Phi(x, f(x)) ~ Phi(x, f(x/(log x)^{k+1})), justified by the claim that log f(x)/log f(x/(log x)^{k+1}) -> 1. That reduces to r(x)/r(x/(log x)^{k+1}) -> 1. The stated hypotheses, r decreasing to 0 with r(x) >>_A (log log x)^A / log x for every A, do not force this. You can construct a decreasing r with arbitrarily large relative drops over the short interval from x/(log x)^{k+1} to x while still respecting the lower bound. The stress-test's specific example, r(x)=exp(-(log log x)^2), actually violates the lower bound, but the point survives: take something like r(x)=exp(sqrt(log log x))/log x and insert sharp downward steps. Then the ratio r(x)/r(x/(log x)^{k+1}) stays bounded away from 1, and the squeeze between the two triangular arrays compares different normalizations. This is load-bearing, not cosmetic.\n\nMinor issues: the Khintchine theorem is quoted with \"monotonically increasing\" where the standard statement uses decreasing; that is fixable. And Theorem 2's sequence version inherits the Section 5.3 gap.\n\nIf the author adds a mild regularity condition (e.g., r(x)/r(x/(log x)^{k+1}) -> 1) or proves it under the current assumptions, the proof likely goes through. The triangular array result alone may already be publishable. I would send this to a serious referee and ask for that repair.","headline":"The paper has a genuinely new idea and a plausible triangular-array theorem, but the sequence-level main theorem rests on an unproved regularity assertion in Section 5.3.","tokens_in":31257,"tokens_out":8562,"would_cite":false,"duration_ms":82138,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11K31","11K36","11N35","11J70","11K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Dilating the set of rough numbers by any badly approximable angle produces a sequence with Poissonian correlations of every order, the first explicit sequence of the form $\\{a_n\\alpha\\}$ with any such property, and a matching converse…","keywords":["rough numbers","Poissonian correlations","Poissonian gaps","badly approximable numbers","Bohr sets","Ostrowski expansions","sieve methods","fine-scale statistics"],"falsifier":"Take $r(x)=(\\log\\log x)^A/\\log x$ on most of each very long interval and set $r(x)=\\frac{1}{2}(\\log\\log x)^A/\\log x$ on the final segment $[x/(\\log x)^{k+1},x]$, with interval endpoints growing fast enough to keep $r$ monotone. This $r$ satisfies the theorem's lower bound yet makes $\\log f(x)/\\log f(x/(\\log x)^{k+1})\\to 2$, so the Section 5.3 equivalence $\\Phi(x,f(x))\\sim\\Phi(x,f(x/(\\log x)^{k+1}))$ fails; that would show the sequence-level theorem does not follow from the triangular-array theorem.","tokens_in":30100,"feed_emoji":"🎲","tokens_out":15207,"duration_ms":151572,"temperature":0.7,"pith_summary":"The paper sets out to show that a number-theoretic sequence, the dilates of the rough integers by a badly approximable angle, behaves statistically like independent random points on the circle. Rough integers are those whose smallest prime factor exceeds a slowly shrinking threshold $f(x)=x^{r(x)}$ with $r(x)\\to 0$; the main theorem states that for every badly approximable $\\alpha$, the sequence $(a_n^{(f)}\\alpha \\bmod 1)$ has Poissonian correlations of every order and therefore Poissonian gaps. This is the first known explicit sequence of the form $\\{a_n\\alpha\\}$ with any Poissonian correlation. The paper also proves a converse: for Lebesgue-almost every $\\alpha$, the same sequence fails to have Poissonian pair correlations, which disproves a 2018 conjecture about such sequences. The mechanism combines a sieve-theoretic count of rough-number tuples with a new equidistribution theorem for residue classes inside Bohr sets.","feed_headline":"Rough-number rotations Poissonian for every badly approximable angle","feed_subtitle":"First explicit {a_n α} sequence with Poissonian statistics of every order; the construction fails for almost every α.","key_machinery":"The argument is carried by three objects working in sequence. The first is the rough-number triangular array $\\{n\\le x: P^-(n)>f(x)\\}$ with counting function $\\Phi(x,z)$, approximated by the Fundamental Lemma of sieve theory. The second is the singular series $S_{k-1}(h)=\\prod_{p\\le f(x)}(1-g_h(p)/p)$, where $g_h(p)$ counts collisions among $0,h_1,\\dots,h_{k-1}$ modulo $p$; Lemma 4 turns this product into an asymptotic for the number of rough $k$-tuples with difference vector $h$. The third is the diophantine Bohr set $B(\\alpha,x)=\\{h\\le x:\\{h\\alpha\\}\\in[0,\\rho_x]\\}$, on which the singular series must be averaged; the decisive new input is Theorem 3, which proves that residue classes modulo $d$ are equidistributed inside such Bohr sets, established through Ostrowski expansions and a Markov-chain convergence argument. A step-function procedure in Section 5.2 converts the averaged Euler products into the rectangle volume $\\mathrm{Vol}(R)$, and Section 5.3 passes from the triangular array to the actual sequence.","core_discovery":"The central claim is Theorem 1: for $f(x)=x^{r(x)}$ with $r(x)$ decreasing monotonically to $0$ and $r(x) \\gg_A (\\log\\log x)^A/\\log x$ for every $A>0$, and for any badly approximable $\\alpha$, the sequence $(a_n^{(f)}\\alpha \\bmod 1)$ has Poissonian correlations of all orders $k\\ge 2$, hence Poissonian gaps, and the same holds for the triangular array of uniformly rough numbers up to $x$. The matching negative result, Theorem 2, says that if $\\alpha$ satisfies $\\liminf_{n\\to\\infty} n\\|n\\alpha\\|\\log f(n)=0$, then the sequence does not even have Poissonian pair correlations; since that Diophantine condition holds for Lebesgue-almost every $\\alpha$, the positive result genuinely depends on bad approximability. The two theorems together refute a 2018 conjecture predicting that non-Poissonian behavior for almost all $\\alpha$ would rule out any Poissonian $\\alpha$.","pith_inferences":["Repairing the Section 5.3 regularity gap would make the method a general template: any sieve-amenable integer set whose singular series averages over a Bohr set should inherit Poissonian correlations for badly approximable $\\alpha$, and the paper already notes one can sift only primes in chosen congruence classes without changing the proof.","The theorem sharpens the expected dichotomy: for a fixed slowly growing sequence, the set of $\\alpha$ with Poissonian correlations can be a measure-zero, full-Hausdorff-dimension set, so the interesting question is not 'almost all $\\alpha$' but which Diophantine null sets work.","A direct numerical test is available: for the golden ratio $\\alpha=(\\sqrt5-1)/2$ and a roughness exponent like $r(x)=1/\\sqrt{\\log x}$, the $k$-point correlations of the rough-number rotation should converge to the rectangle volume, though the implied convergence has no rate."],"forward_implications":["For every badly approximable $\\alpha$, the rough-number rotation has Poissonian correlations of every order and therefore Poissonian gaps, giving the first known explicit sequence of the form $\\{a_n\\alpha\\}$ with any Poissonian correlation.","The gap distribution between successive points is exponential, and for every fixed $k$ the distribution of $k$-th neighbour spacings is determined by the same correlation functions.","For Lebesgue-almost every $\\alpha$, the same sequence fails even to have Poissonian pair correlations, so the positive result cannot be extended to a metric statement; this refutes a 2018 conjecture claiming that non-Poissonian behavior for almost all $\\alpha$ would force non-Poissonian behavior for every $\\alpha$.","The equidistribution theorem for residue classes inside Bohr sets is flexible enough to average multiplicative functions over Bohr sets, which the paper identifies as a tool of independent interest."],"supporting_citations":[{"why":"Gives the pair-correlation conjecture for polynomial rotations that motivates the paper, and supplies the badly approximable hypothesis.","marker":"[47]"},{"why":"Provides the Fundamental Lemma of sieve theory used to count rough $k$-tuples with a fixed difference vector.","marker":"[26]"},{"why":"Develops the real Ostrowski expansion and cylinder sets on which the Bohr-set equidistribution proof rests.","marker":"[6]"},{"why":"Provides the structural inclusion of Bohr sets in small boxes used to bound residue-class counts in Proposition 12.","marker":"[12]"},{"why":"Supplies the template for the non-Poissonian converse: approximants of $\\alpha$ create too many Bohr-set elements.","marker":"[58]"},{"why":"States the conjecture about pair correlations of $\\{a_n\\alpha\\}$ that the paper disproves.","marker":"[31]"},{"why":"Prior work giving full Poissonian statistics for slowly growing real sequences, the contrast class for the current result.","marker":"[35]"},{"why":"Contains the implication from Poissonian correlations of all orders to Poissonian gaps used for the final conclusion.","marker":"[29]"}],"fun_headline_variants":["Rough rotations Poissonian for all bad angles","First explicit Poissonian sequence for every bad angle","Badly approximable angles force Poissonian rough rotations","Rough-number gaps Poissonian: conjecture disproved","Poissonian every order, but only for bad angles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The transfer from the uniformly rough triangular array to the true sequence assumes that replacing the cutoff $f(x)$ by $f(x/(\\log x)^{k+1})$ changes neither the counting function nor the correlation limit, because the two cutoffs are treated as asymptotic powers of one another; the stated hypotheses on decreasing $r(x)$ do not force that ratio $\\log f(x/(\\log x)^{k+1})/\\log f(x)$ to tend to $1$.","fun_headline_variants_meta":{"raw":{"variants":["Rough rotations Poissonian for all bad angles","First explicit Poissonian sequence for every bad angle","Badly approximable angles force Poissonian rough rotations","Rough-number gaps Poissonian: conjecture disproved","Poissonian every order, but only for bad angles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000154,"raw_usage":{"total_tokens":1227,"prompt_tokens":981,"completion_tokens":246,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":171}},"tokens_in":597,"tokens_out":246,"duration_ms":3017,"temperature":1.0,"reasoning_tokens":171,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:39:42.157557+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $r(x)=(\\log\\log x)^A/\\log x$ on most of each very long interval and set $r(x)=\\frac{1}{2}(\\log\\log x)^A/\\log x$ on the final segment $[x/(\\log x)^{k+1},x]$, with interval endpoints growing fast enough to keep $r$ monotone. This $r$ satisfies the theorem's lower bound yet makes $\\log f(x)/\\log f(x/(\\log x)^{k+1})\\to 2$, so the Section 5.3 equivalence $\\Phi(x,f(x))\\sim\\Phi(x,f(x/(\\log x)^{k+1}))$ fails; that would show the sequence-level theorem does not follow from the triangular-array theorem.","supporting_citations":[{"cited_title":"Rudnick and P","cited_arxiv_id":null,"evidence_quote":"Gives the pair-correlation conjecture for polynomial rotations that motivates the paper, and supplies the badly approximable hypothesis."},{"cited_title":"Koukoulopoulos.The distribution of prime numbers, volume 203 ofGraduate Studies in Mathematics","cited_arxiv_id":null,"evidence_quote":"Provides the Fundamental Lemma of sieve theory used to count rough $k$-tuples with a fixed difference vector."},{"cited_title":"Beresnevich, A","cited_arxiv_id":null,"evidence_quote":"Develops the real Ostrowski expansion and cylinder sets on which the Bohr-set equidistribution proof rests."},{"cited_title":"Chow and N","cited_arxiv_id":null,"evidence_quote":"Provides the structural inclusion of Bohr sets in small boxes used to bound residue-class counts in Proposition 12."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the template for the non-Poissonian converse: approximants of $\\alpha$ create too many Bohr-set elements."},{"cited_title":"Larcher and W","cited_arxiv_id":null,"evidence_quote":"States the conjecture about pair correlations of $\\{a_n\\alpha\\}$ that the paper disproves."},{"cited_title":"Lutsko and N","cited_arxiv_id":null,"evidence_quote":"Prior work giving full Poissonian statistics for slowly growing real sequences, the contrast class for the current result."},{"cited_title":"Kurlberg and Z","cited_arxiv_id":null,"evidence_quote":"Contains the implication from Poissonian correlations of all orders to Poissonian gaps used for the final conclusion."}],"review_version":1}