REVIEW 2 major objections 4 minor 28 references
Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\{n\alpha\}$
T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The paper proves a phase transition at $\beta\gamma=2$ in the expected $p$-Wasserstein distance of random-walk empirical measures on the circle, with rates $n^{-1/2}$, $(\log n)^{1-1/(p\vee 2)}n^{-1/2}$, and $n^{-1/(\beta\gamma)}$.
desk verdict A serious, mostly solid paper that likely settles Wasserstein rates for random subsequences of {nα}, but the sharpest lower bound rests on a load-bearing omitted proof (Lemma 5.7) that must be supplied before the strongest claim can be accepted. 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 four interlocking objects. First, the antiderivative $F_\nu(x)=\nu([0,x))-x$ converts $W_p(\nu,\text{uniform})$ into the $L^p$ norm of $F_\nu$ up to a shift, so Fourier coefficients $\hat\nu(m)/(m)$ control the Wasserstein distance. Second, the heat semigroup $P_t=e^{t\Delta}$ smooths $\mu_n$ and produces a Fourier-weighted estimate (Lemma 3.2) whose optimization over $t$ yields the switch between $n^{-1/2}$ and $n^{-1/(\beta\gamma)}$. Third, continued-fraction convergents $p_k/q_k$ of $\alpha$ convert the Diophantine hypothesis into sharp bounds on sums such as $\sum_{q_k\le m<q_{k+1}} m^{-\theta}\|m\alpha\|^{-\tau}$ (Lemma 3.5), which control the tail of the Fourier sums. Fourth, for lower bounds, a geometric lemma states that if a measure avoids $K$ disjoint intervals of lengths $L_i$ on the circle, then $W_1(\nu,\text{uniform})\ge \sum_i L_i^2/4$; the near-lattice clustering of $\{j\alpha:|j|\le K(q)\}$ then yields the $n^{-1/(\beta\gamma)}$ lower bound. The sharp $\sqrt{\log n/n}$ rate for badly approximable $\alpha$ comes from a new moment estimate for the negative-Sobolev norm of the smoothed density (Proposition 5.5), proved by decomposing the $2p$-th moment into blocks of vanishing Fourier frequencies.
What would settle it
Take $\alpha=(\sqrt5-1)/2$, the golden ratio, which is badly approximable, and take $X_i$ to be a simple $\pm1$ random walk, so $\beta=2$, zero mean, and finite variance. Theorem 1.5 then predicts $\mathbb{E}[W_2^2(\mu_n,\mu)]\asymp \log n/n$. A direct Monte Carlo estimate of $\mathbb{E}[W_2^2(\mu_n,\mu)]$ for $n=10^3,10^4,10^5,10^6$, with enough independent walks to control sampling error, should keep $n(\log n)^{-1}\mathbb{E}[W_2^2]$ bounded between two positive constants; if it instead decays to $0$ or diverges, the claimed rate fails.
Extended reading notes
Core claim
On the paper's own terms, the discovery is Theorems 1.4 and 1.5. Fix an irrational $\alpha$ satisfying $0<\liminf_{q\to\infty} q^\gamma\|q\alpha\|<\infty$, and let $X_i$ be integer-valued i.i.d. random variables whose characteristic function obeys $|1-\varphi(x)|\asymp |x|^\beta$ near $0$. Then for the empirical measure $\mu_n=\frac1n\sum_{j=1}^n\delta_{\{S_j\alpha\}}$ and every $1\le p<\infty$, $$\mathbb{E}[W_p(\mu_n,\mu)]\asymp $n^{{-1/2}}$\quad(\$\beta$\gamma<2),$$ $$c_1 $n^{{-1/2}}$\le \mathbb{E}[W_p(\mu_n,\mu)]\le c_2(\log n)^{1-\frac{1}{p\vee 2}}\,$n^{{-1/2}}$\quad(\$\beta$\gamma=2),$$ $$0<\limsup_{n\to\infty} $n^{{1/(\beta\gamma)}}$\,\mathbb{E}[W_p(\mu_n,\mu)]<\infty\quad(\$\beta$\gamma>2).$$ For badly approximable $\alpha$ (so $\gamma=1$) and steps with zero mean and finite variance (so $\beta=2$), the paper proves $\mathbb{E}[W_p(\mu_n,\mu)]\asymp\sqrt{\log n/n}$ for $1\le p\le 2$. The same machinery yields almost-sure limsup lower bounds for $W_1$.
Load-bearing premise
The load-bearing premise is the two-sided Diophantine condition $0<\liminf_{q\to\infty} q^\gamma\|q\alpha\|<\infty$, together with matching upper and lower power decay $|1-\varphi(x)|\asymp |x|^\beta$ near $0$; if either side fails, the phase transition and the claimed rates are not established, and for $\gamma>1$ the admissible rotations form an uncountable but Lebesgue-null set.
Editorial extensions
If this is right
- For $\beta\gamma<2$, any non-constant integer-valued walk has $\mathbb{E}[W_p(\mu_n,\mu)]\asymp n^{-1/2}$ for every $1\le p<\infty$, and for $p=2$ the normalized limit $\lim_{n\to\infty} n\mathbb{E}[W_2^2(\mu_n,\mu)]$ is given by the explicit Fourier sum in Proposition 4.4.
- At $\beta\gamma=2$ the paper leaves a logarithmic gap in general, with lower order $n^{-1/2}$ and upper order $(\log n)^{1-1/(p\vee2)}n^{-1/2}$, but closes it to $\sqrt{\log n/n}$ for badly approximable $\alpha$ with finite-variance zero-mean steps.
- For $\beta\gamma>2$, convergence is slower, with $0<\limsup_{n\to\infty} n^{1/(\beta\gamma)}\mathbb{E}[W_p(\mu_n,\mu)]<\infty$, and the same order appears as an almost-sure limsup lower bound for $W_1$.
- Pathwise lower bounds follow in all regimes: $\sqrt{n/\log\log n}\,W_1$ has a positive limsup when $\beta\gamma<2$, and $n^{1/(\beta\gamma)}W_1$ has a positive limsup when $\beta\gamma>2$, with the first known to be optimal against matching upper bounds.
Reading between the lines
- Inference: for $\beta\gamma<2$, the rate's independence of $p$ and of the finer Diophantine structure suggests a distributional convergence theorem for $W_p$ around a Gaussian limit, with the Fourier sum in Proposition 4.4 as the natural covariance candidate; the paper does not establish such a limit.
- Inference: at the critical line $\beta\gamma=2$, the gap between the general upper bound and the lower bound is probably genuine for large $p$, and the exact $p$-dependence of the logarithmic correction is left open; a natural test is to compute the rate for simple random walks on a golden-ratio rotation.
- Inference: because the two-sided Diophantine condition is Lebesgue-null for $\gamma>1$, the theorem does not cover typical rotations; averaging over $\alpha$ or over the step distribution would be a natural way to see whether the phase transition survives for generic parameters.
- Inference: the same heat-semigroup plus continued-fraction scheme should extend to integer lattice walks on higher-dimensional tori, with a threshold depending on dimension, Diophantine type, and step characteristic exponent; this is a testable extension rather than a claim of the paper.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the empirical measure μ_n = (1/n)∑_{j=1}^n δ_{S_j α} on the torus, where S_j is a random walk with integer-valued i.i.d. increments and α is irrational. Under a lower Diophantine assumption on α and a Hölder-type condition |1−φ(x)| ≍ |x|^β on the characteristic function, it proves upper bounds for the expected p-Wasserstein distance to the uniform measure, exhibiting a phase transition at βγ = 2 (Theorem 1.1). It complements these with a universal W_1 lower bound of order n^{−1/2} (Theorem 1.2) and a lower bound of order n^{−1/(βγ)} along a subsequence when an upper Diophantine condition holds (Theorem 1.3), yielding the combined rates in Theorem 1.4 under a two-sided condition. For badly approximable α and finite-variance zero-mean steps, Theorem 1.5 gives the sharp expected rate √(log n/n) for 1 ≤ p ≤ 2, together with pathwise lower-bound corollaries. The proofs use heat-semigroup smoothing, Fourier/Erdős–Turán-type estimates, and continued-fraction block estimates.
Significance. The main results, if correct, settle the expected Wasserstein convergence rates for random subsequences of Kronecker sequences and identify the critical product βγ = 2. The upper-bound derivation in Section 4 is carefully written, with explicit parameter tracking and a clean use of the continued-fraction block estimate (Lemma 3.5). The lower-bound strategy is also attractive: Lemma 5.2 gives a general L1-Wasserstein/lower-bound device via Kantorovich duality, and its application in Lemma 5.4 to Diophantine rational approximations is elegant. The sharp logarithmic factor in Theorem 1.5 is a natural target and is supported by a plausible moment-estimation framework. The paper also provides several pathwise corollaries that go beyond the expected-value statements. Overall, this is a substantial contribution to the quantitative ergodic theory of random walks on the torus, complementing the discrepancy results of Berkes and Borda.
major comments (2)
- [Section 5.3.1, Lemma 5.7] The proof of Lemma 5.7 is omitted, with the note 'The proof of the first one simply uses Lemma 3.5 and the argument presented in Section 4.1 so we omit it.' This lemma is genuinely load-bearing: it is used in the proof of Lemma 5.8 to control the R1-sum, and Lemma 5.8 is then used inductively in Lemma 5.9 and Proposition 5.5, which supplies the B(n,n^{-1/2}) bound required for the lower bound in Theorem 1.5. The stated bound O(log ε^{-1}) is plausible but not immediate: while Lemma 3.5 gives per-block estimates whose leading term is O(q_k^{-θ} ||q_k α||^{-τ}) = O(1) under θ = γτ, the passage from block sums to the full series with the Gaussian cutoff e^{-m^2 ε} requires a written argument controlling the number of blocks and the tail. The authors should supply a complete proof or a precise reference; in its current form the sharp lower bound of Theorem 1.5 is not verified as written.
- [Section 5.3, Eq. (5.8)] The lower bound for E[W_1(μ_n, μ)] in the proof of Theorem 1.5 is obtained from an inequality attributed to [14, Lemma 2.1], but that lemma is not stated in the paper. Since the subsequent choice M = Θ√(log n/n) and the dependence of the term B(n,ε) on M^3 are delicate, the authors should state the precise form of the inequality used, including any constants and the range of ε for which it holds, or give a self-contained proof. This is a checkable but currently missing step in the critical-case argument.
minor comments (4)
- [Section 5.3.1, proof of Lemma 5.8] In the displayed estimate for the R1-sum, the factor e^{-2π²(m1−m)²ε} appears twice in the numerator; one factor should be absorbed into the bound for |m1−m| e^{−2π²(m1−m)²ε}, so the display should be corrected to avoid a duplicated exponential.
- [Section 5.2, proof of Theorem 1.3] The formula for N(q) should be typeset as ⌊q^{βγ}/(3CC_1)^β⌋ so that the exponent β applies to the whole denominator; the current typesetting 'qβγ/(3CC 1)β' is ambiguous and could be misread as q^{βγ}/(3CC_1^β), which would break the required inequality C_1 N(q)^{1/β} ≤ q^γ/(3C).
- [Section 5.3.1, Proposition 5.5] In the final display of the proof, the absorption of the ε^{-1/2} log(ε^{-1}) factor into ε^{-1} uses an inequality that is valid for ε ∈ (0,1/2), but the inequality is not written; adding a one-line justification would improve readability.
- [Section 4.1, end of proof of Theorem 1.1] The choice ε = κ n^{−2/(βγ∨2)} should specify that κ > 0 is chosen sufficiently small so that the condition ε < 1/2 (and, in the critical case, the condition ε ≥ κ n^{−1/2} used later) is satisfied; this is implicit but should be stated.
Circularity Check
No significant circularity: the Wasserstein rate theorems are derived from the Diophantine and characteristic-function hypotheses via independent lemmas, with no fitted parameters or assumption of the target rates.
full rationale
The paper's derivation chain for Theorems 1.1–1.5 does not assume the target rates as inputs. Upper bounds are obtained from Lemma 3.2 (Fourier/Wasserstein comparison), Lemma 4.1 (variance bounds on Fourier coefficients from |1−φ(x)| ≥ c|x|^β), and Lemma 3.5 (continued-fraction block estimates), with no parameter fitted to the output. Lower bounds are obtained from Kantorovich duality and the new lower-bound Lemma 5.2, whose conclusion depends only on the size of excluded intervals, not on any Wasserstein rate. Theorem 1.4 is a direct conjunction of Theorems 1.1–1.3. Theorem 1.5's sharp rate depends on Proposition 5.5 and Lemmas 5.6–5.9, which estimate moments of the smoothed empirical measure from the structural hypotheses; none of these lemmas assumes the ≍ √(log n/n) conclusion. The self-citations [25] and [26] appear only in the list of related PDE-approach works and are not used to justify any load-bearing step. The omitted proof of Lemma 5.7 is an internal completeness gap, not a circularity: the lemma is claimed to follow from Lemma 3.5 and the Section 4.1 argument, and even if that proof were missing, that would be unverified support, not an equation reducing to its own output. No fitted-input-called-prediction, no uniqueness imported from authors, and no ansatz smuggled in via citation are present.
Assumptions & free parameters
assumptions (5)
- domain assumption The two-sided Diophantine condition 0 < liminf_{q→∞} q^γ ‖qα‖ < ∞ (Eq. (1.3)).
- domain assumption Two-sided power behavior of the characteristic function: |1-φ(x)| ≍ |x|^β in a neighborhood of 0 (condition (i) of Theorem 1.4).
- domain assumption In Theorem 1.5: EX_1=0 and 0<EX_1^2<∞, so β=2.
- standard math External published lemmas: [5, Proposition 2.2] (moments of random exponential sums), [14, Lemma 2.1] (W1 lower bound via H^{-1,p} norms), [4, Lemma 6.2] (moderate deviation bound on M_n).
- standard math Standard background: Weyl criterion, Dirichlet theorem, Roth theorem, Jarník-Besicovitch theorem, Hausdorff-Young inequality, Kantorovich duality, Hewitt-Savage zero-one law.
Cite this review
Pith. "Pith review of Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\{n\alpha\}$." pith.science (2026). https://pith.science/paper/3Q2JYZNN
@misc{pith2026241115724,
author = {Pith},
title = {Pith review of: Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\n\alpha\$},
year = {2026},
howpublished = {\url{https://pith.science/paper/3Q2JYZNN}},
note = {Machine review of arXiv:2411.15724}
}
abstract
Fix an irrational number $\alpha$. Let $X_1,X_2,\cdots$ be independent, identically distributed, integer-valued random variables with characteristic function $\varphi$, and let $S_n=\sum_{i=1}^n X_i$ be the partial sums. Consider the random walk $\{S_n \alpha\}_{n\ge 1}$ on the torus, where $\{\cdot\}$ denotes the fractional part. We study the long time asymptotic behaviour of the empirical measure of this random walk to the uniform distribution under the general $p$-Wasserstein distance. Our results show that the Wasserstein convergence rate depends on the Diophantine properties of $\alpha$ and the H\"older continuity of the characteristic function $\varphi$ at the origin, and there is an interesting critical phenomenon that will occur. The proof is based on the PDE approach developed by L. Ambrosio, F. Stra and D. Trevisan in [2] and the continued fraction representation of the irrational number $\alpha$.
Reference graph
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Reviewed August 12, 2026 · model on record in the stance chip above.
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