{"id":"80a3b044-3dae-4422-880c-a4ffc9560ce9","arxiv_id":"2411.15724","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For random subsequences of the rotation {nα}, the expected p-Wasserstein distance of the empirical measure to uniformity decays like n^{-1/2}, (log n)^{1-1/(p∨2)} n^{-1/2}, or n^{-1/(βγ)} depending on whether βγ<2, =2, or >2.","lead":"This paper determines the rate at which the empirical distribution of a random walk on the circle approaches the uniform distribution in Wasserstein distance. The rate depends on the Diophantine type of the rotation angle and the tail behavior of the random step, with a sharp phase transition at a critical parameter value.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5's sharp lower bound depends on Lemma 5.7, whose proof is omitted; the claimed O(log ε^{-1}) in that lemma is not obvious from Lemma 3.5 and must be supplied.","rationale":"The reader's verdict is CONDITIONAL and already flags Lemma 5.7 in the rationale, but the stated weakest_assumption emphasizes the two-sided Diophantine condition (1.3). I agree that (1.3) limits the theorem to a measure-zero class of α, yet that is a scope condition, not an internal inconsistency: the paper explicitly restricts itself to W(γ) and proves the relevant cardinality statement in Remark 3.4. The load-bearing issue is instead the omitted proof of Lemma 5.7, which is needed for Proposition 5.5 and therefore for Theorem 1.5's sharp sqrt(log n/n) lower bound. I checked the surrounding chain: Lemma 5.8 and Lemma 5.9 are written in enough detail that they can be followed, and the final choice M=Θ√(log n/n) in the application of [14, Lemma 2.1] is consistent, though it relies on an external inequality. The main missing piece is Lemma 5.7. My proposed test combines an analytical reconstruction of the lemma with a numerical sanity check for a badly approximable α; this would settle whether the sharp lower bound is valid. No change to the reader's CONDITIONAL verdict is needed, since the reader already required revision; my concern reinforces that requirement with a specific, located gap.","tokens_in":68,"tokens_out":28699,"duration_ms":371007,"concrete_test":"Supply a complete proof of Lemma 5.7 by splitting the sum into convergents blocks q_k≤m<q_{k+1}, applying Lemma 3.5 blockwise, and summing with the weight e^{-m²ε}, verifying that the leading term is O(1) because θ=γτ and that the total over k is O(log ε^{-1}) plus a convergent tail. As a numerical cross-check, compute S(ε)=∑_{m=1}^{10000} e^{-m²ε}/(m²||mα||²) for α=(√5-1)/2 (golden ratio, γ=1, θ=τ=2) at ε=10^{-3},10^{-4},10^{-5},10^{-6}; if S(ε) grows like C log(1/ε) the lemma is consistent, whereas growth like C ε^{-1/2} would falsify it and break the proof of Theorem 1.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The sharp rate E[W_p(μ_n,μ)] ≍ sqrt(log n/n) in Theorem 1.5 is the strongest advertised result. Its lower bound passes through Proposition 5.5, whose proof uses Lemma 5.9, which in turn relies on Lemma 5.8 and Lemma 5.7. Lemma 5.7 is explicitly stated without proof: the text says 'The proof of the first one simply uses Lemma 3.5 and the argument presented in Section 4.1 so we omit it.' This is a genuine load-bearing gap: the lemma asserts that, under θ=γτ, ∑_{m≠0} e^{-m²ε}/(|m|^θ||mα||^τ) = O(log ε^{-1}). The omitted step is the interaction of the Gaussian cutoff e^{-m²ε} with the continued-fraction block estimates of Lemma 3.5. The leading block term in Lemma 3.5 is O(q_k^{γτ-θ}) = O(1) precisely because θ=γτ, so the block sums are bounded uniformly; summing over the O(log ε^{-1}) blocks with q_k ≲ ε^{-1/2} plausibly gives the stated log divergence. But this deduction is not written, and a different treatment of the tail could introduce an extra ε^{-1/2} factor. If Lemma 5.7 fails, Lemma 5.8's bound becomes too weak, Lemma 5.9's induction produces F(N,ε)=O(ε^{-(N-1)} log ε^{-1}) instead of O(ε^{-(N-2)} log ε^{-1}), and Proposition 5.5 no longer gives B(n,n^{-1/2}) ≤ C(log n)^2/n². Consequently the lower bound in Theorem 1.5 is unverified as written. An independent proof of Lemma 5.7, or a reference containing it, is required before the sharp critical-case rate can be accepted. This is not a documented mathematical error, but it is the single most load-bearing unverified step in the strongest claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25369,"tokens_out":17859,"duration_ms":149085,"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":[{"comment":"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":"Section 5.3.1, Lemma 5.7"},{"comment":"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.","section":"Section 5.3, Eq. (5.8)"}],"minor_comments":[{"comment":"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":"Section 5.3.1, proof of Lemma 5.8"},{"comment":"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":"Section 5.2, proof of Theorem 1.3"},{"comment":"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":"Section 5.3.1, Proposition 5.5"},{"comment":"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.","section":"Section 4.1, end of proof of Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper self-cites [25] and [26] only as related work, and no step of the proof appears to rely on those papers to force a conclusion; I did not detect citation inflation. The restriction of Theorem 1.4 to the set W(γ), which has Lebesgue measure zero when γ > 1, is inherent to the phase-transition claim and is stated clearly, so I do not treat it as a defect. Once the omitted proof of Lemma 5.7 is supplied (and the auxiliary inequality (5.8) is either stated or made verifiable), the paper should be suitable for publication in a serious probability journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the paper carefully. The main results are likely correct, and the upper bound half is genuinely well executed. The contribution: for random subsequences of {nα}, the paper gives the first complete set of p-Wasserstein rates for all 1≤p<∞, including the phase transition at βγ=2 and matching lower bounds for the subcritical and supercritical regimes. Theorem 1.4(1) and (3) are established by combining Theorem 1.1 with Theorems 1.2 and 1.3. The upper bound proof is internally consistent; Lemma 4.1 and the continued-fraction block estimate in Lemma 3.5 are properly used. The new moment estimation technique in Section 5.3.1 is a real technical innovation, even if its presentation is terse.\n\nThe soft spot is exactly where the stress-test note puts it: Lemma 5.7. Its proof is omitted, just 'we omit it,' and the lemma is load-bearing for the sharp lower bound in Theorem 1.5. The lemma asserts an O(log ε^{-1}) bound for a sum mixing a Gaussian cutoff with Diophantine block estimates. The leading block term via Lemma 3.5 is O(1), and summing over O(log ε^{-1}) blocks plausibly gives the bound, but the tail behavior needs care. I share the stress-test's conclusion: this deduction is not written, and without it Theorem 1.5 is unverified as is. This is not a documented mathematical error; the gap looks fillable. But the strongest advertised result should not be accepted without the proof or a reference.\n\nMinor concerns: the p=2 lower bound in the critical case also relies on [14, Lemma 2.1], an external PDE inequality; that is acceptable but worth flagging. The two-sided Diophantine condition (1.3) is restrictive (Lebesgue-null for γ>1), but that is built into the sharp statement, and the paper says so. The citation pattern is honest; the self-cited works appear only as related work, not as proof dependencies.\n\nWho this is for: specialists in Diophantine approximation and empirical measure asymptotics. The paper deserves a serious referee. I'd send it to peer review with the explicit request to verify Lemma 5.7 and the use of [14, Lemma 2.1]. If those check out, this is a strong contribution.","headline":"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.","tokens_in":25983,"tokens_out":2244,"would_cite":true,"duration_ms":20227,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G50","60B10","11J70","42A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $n^{-1/(\\beta\\gamma)}$.","keywords":["Wasserstein distance","empirical measure","random walk on the torus","Diophantine approximation","continued fractions","characteristic function","Kronecker sequence","phase transition"],"falsifier":"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.","tokens_in":24745,"feed_emoji":"🎲","tokens_out":19041,"duration_ms":148053,"temperature":0.7,"pith_summary":"This paper asks how fast the empirical measure of a random subsequence of the Kronecker sequence $\\{j\\alpha\\}$ approaches the uniform measure on the circle, measured in $p$-Wasserstein distance. The subsequence is generated by partial sums $S_j=X_1+\\cdots+X_j$ of integer-valued i.i.d. steps, giving points $\\{S_j\\alpha\\}$ on the torus. The central claim is a phase transition: if $\\beta$ is the power exponent in $|1-\\varphi(x)|\\asymp |x|^\\beta$ near the origin and $\\gamma$ is the Diophantine type of $\\alpha$, then the expected $p$-Wasserstein distance is of order $n^{-1/2}$ when $\\beta\\gamma<2$, at most a logarithmic factor above $n^{-1/2}$ when $\\beta\\gamma=2$, and of order $n^{-1/(\\beta\\gamma)}$ when $\\beta\\gamma>2$. For badly approximable $\\alpha$ with finite-variance zero-mean steps, the paper proves the exact order $\\sqrt{\\log n/n}$. If correct, this settles the Wasserstein convergence rates for random subsequences of $\\{n\\alpha\\}$ and identifies the critical product $\\beta\\gamma=2$ that separates fast from slow convergence.","feed_headline":"Phase transition at βγ=2 in random-walk circle measures","feed_subtitle":"Below βγ=2 convergence is parametric; at the threshold a log factor appears; above it the rate slows to n^{-1/(βγ)}.","key_machinery":"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.","core_discovery":"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 $W_1$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the PDE and heat-semigroup smoothing method that converts Wasserstein estimates into Fourier-coefficient estimates.","marker":"[2]"},{"why":"proves the antiderivative representation of $W_p$ and the $L^p$ Fourier-weighted inequality used for the upper bounds.","marker":"[13]"},{"why":"established the analogous discrepancy phase transition for random subsequences and supplied the continued-fraction sum estimate and walk-concentration bound used in the lower bounds.","marker":"[4]"},{"why":"provides moment estimates and the law of the iterated logarithm for random exponential sums used in Theorem 1.2 and Corollary 1.8.","marker":"[5]"},{"why":"gives the negative-Sobolev moment bound and the lower-bound inequality that the proof of Theorem 1.5 refines.","marker":"[14]"},{"why":"supplies the definition of $W_p$, Kantorovich duality, and joint convexity used throughout the upper and lower bound arguments.","marker":"[23]"}],"fun_headline_variants":["Phase transition at βγ=2 in random-walk measures","Random walk on circle: Wasserstein rate toggles at βγ=2","Empirical measures of nα subsequence: rate switches at threshold","Wasserstein convergence for random subsequences: critical βγ=2","Diophantine critical point controls empirical measure rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Phase transition at βγ=2 in random-walk measures","Random walk on circle: Wasserstein rate toggles at βγ=2","Empirical measures of nα subsequence: rate switches at threshold","Wasserstein convergence for random subsequences: critical βγ=2","Diophantine critical point controls empirical measure rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000656,"raw_usage":{"total_tokens":3068,"prompt_tokens":1071,"completion_tokens":1997,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":1907}},"tokens_in":687,"tokens_out":1997,"duration_ms":12797,"temperature":1.0,"reasoning_tokens":1907,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:03:39.677274+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ambrosio, F","cited_arxiv_id":null,"evidence_quote":"supplies the PDE and heat-semigroup smoothing method that converts Wasserstein estimates into Fourier-coefficient estimates."},{"cited_title":"Graham, Irregularity of distribution in Wasserstein distance","cited_arxiv_id":null,"evidence_quote":"proves the antiderivative representation of $W_p$ and the $L^p$ Fourier-weighted inequality used for the upper bounds."},{"cited_title":"Berkes, B","cited_arxiv_id":null,"evidence_quote":"established the analogous discrepancy phase transition for random subsequences and supplied the continued-fraction sum estimate and walk-concentration bound used in the lower bounds."},{"cited_title":"Berkes, B","cited_arxiv_id":null,"evidence_quote":"provides moment estimates and the law of the iterated logarithm for random exponential sums used in Theorem 1.2 and Corollary 1.8."},{"cited_title":"Huesmann, F","cited_arxiv_id":null,"evidence_quote":"gives the negative-Sobolev moment bound and the lower-bound inequality that the proof of Theorem 1.5 refines."},{"cited_title":"Villani, Topics in Optimal Transportation , Graduate Studies in Mathematics, vol","cited_arxiv_id":null,"evidence_quote":"supplies the definition of $W_p$, Kantorovich duality, and joint convexity used throughout the upper and lower bound arguments."}],"review_version":1}