REVIEW 2 major objections 5 minor 1 cited by
The bad and rough rotation is Poissonian
T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read 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…
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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 $\alpha$.
Load-bearing premise
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 $1$.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Section 5.3] 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 6] 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.
minor comments (5)
- [Definition 1] The definition of Φ(x,z) reads '#{1≤n≤x : P^−(x)>z}'; the argument of P^− should be n, not x.
- [Lemma 11] 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.
- [Notation, Section 2.0.1] 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 5.3] 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.
- [Proposition 13] 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.
Circularity Check
No significant circularity: the Poissonian correlation result is derived from independent sieve estimates, a new Bohr-set equidistribution theorem, and elementary averaging arguments; the self-citations are not load-bearing in a circular way.
full rationale
The paper's central claim in Theorem 1 is not obtained by fitting a parameter and then renaming it as a prediction, nor by defining an object in terms of the target conclusion. The proof proceeds through a uniform sieve estimate (Lemma 4), a new equidistribution result modulo d in badly approximable Bohr sets (Theorem 3, proved via Ostrowski expansions and Markov chains), an averaging argument over multiplicative functions (Lemma 16), and a step-function/Riemann-sum approximation. At no point does the proof presuppose the Poissonian value Vol(R) that it derives; the limit emerges only after taking N to infinity and then M to infinity in the Euler-product argument. The two self-citations are to [20] for a Bohr-set counting bound used in tail estimates and to [21] for an elementary integral evaluation; neither imports a uniqueness theorem, neither defines the rough-number sequence in terms of the correlation function, and neither has assumptions containing the target Poissonian statement. Under the stated rules these are independent supports rather than circular ones. There is a potential analytic gap in Section 5.3, where the transfer from triangular arrays to the actual sequence uses 'Φ(x, z) ∼ Φ(x, z−) since log z / log z− → 1'; the stated monotonicity and lower-bound assumptions on r(x) may not force that ratio to tend to 1. This is a correctness/falsifiability concern and would need a regularity condition such as r(x)/r(x/(log x)^(k+1)) → 1, but it is not circularity: the contested ratio is not an input disguised as the output, and no equation in the chain is equivalent by construction to the claimed correlation limit. Therefore the circularity score is 0.
Assumptions & free parameters
assumptions (7)
- standard math Fundamental Lemma of Sieve Theory as stated in Lemma 5
- standard math Denjoy-Koksma inequality for rotations
- standard math Ostrowski expansion existence, uniqueness, and cylinder-set correspondence
- standard math Markov chain convergence theorem for finite irreducible aperiodic chains
- standard math Khintchine's theorem in metric Diophantine approximation
- domain assumption Alpha is badly approximable, so its partial quotients are bounded
- domain assumption Roughness function r(x) decreases to 0 and r(x) >>_A (log log x)^A / log x for every A
Cite this review
Pith. "Pith review of The bad and rough rotation is Poissonian." pith.science (2026). https://pith.science/paper/G42E6XCF
@misc{pith2026250601736,
author = {Pith},
title = {Pith review of: The bad and rough rotation is Poissonian},
year = {2026},
howpublished = {\url{https://pith.science/paper/G42E6XCF}},
note = {Machine review of arXiv:2506.01736}
}
abstract
Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ with $(a_n)_{n \in \mathbb{N}} \subseteq \mathbb{N} $ and $\alpha$ irrational have been extensively studied in the last decades. In this article, we prove that for $(a_n)_{n \in \mathbb{N}}$ arising from the set of rough numbers with explicit roughness parameters and any badly approximable $\alpha$, $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first known explicit sequence $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ with these properties. Further, we show that this result is false for Lebesgue almost every $\alpha$, thereby disproving a conjecture of Larcher and Stockinger [Math. Proc. Camb. Phil. Soc. 2020]. The method of proof makes use of an equidistribution result mod $d$ in diophantine Bohr sets which might be of independent interest.
Forward citations
Cited by 1 Pith paper
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Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution
Weak γ-PPC implies neither γ-PPC nor equidistribution, and weak γ1-PPC does not imply weak γ2-PPC for distinct γ in (0,1/2].
Reviewed August 7, 2026 · model on record in the stance chip above.
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