{"id":"ff3d5c35-318b-4dff-8cb1-ad747b88b8ab","arxiv_id":"2505.22059","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit Wasserstein-distance rates are proved for equidistribution of ultra-short exponential sums and of Deligne-Katz trace-function families over finite fields.","lead":"Wasserstein distance, a statistician's way to measure how far apart two probability distributions are, is turned into the quantitative yardstick for equidistribution of exponential sums over finite fields, with explicit error rates proved in several classical theorems. The results upgrade qualitative equidistribution theorems of Deligne and Katz into rate-controlled bounds and yield new statements, like shrinking targets for hyper-Kloosterman sums.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.11's proof is internally inconsistent: the displayed Fourier bound ||λ||^{c||λ||} is replaced by c^{||λ||} in the Borda sum, so the claimed 1/log q rate is not justified as written.","rationale":"The reader's weakest assumption (connectedness of monodromy) is an explicit hypothesis in Theorem 4.4 and is already flagged in Remark 4.2(1); it limits scope but does not create an internal gap. The Theorem 4.11 inconsistency, by contrast, is a concrete internal gap in the proof of a stated quantitative equidistribution theorem advertised in the abstract as a quantitative form of Katz's theorem. It is checkable: the displayed bound ||λ||^{c||λ||} cannot be replaced by c^{||λ||} in the summation, and the claimed 1/log q rate does not follow as written. The rest of the paper appears sound: I checked the Fourier-vanishing and norm-threshold argument in Section 3, the d = o(log q/log log q) balance in Theorem 3.8, the Borda-sum optimization in Theorem 4.4 for semisimple K, and the reduction of the variety case in Theorem 4.9. Therefore the verdict should move from ACCEPT to CONDITIONAL: the main Theorems 3.1, 3.8, 4.4 and 4.9 stand, but Theorem 4.11's proof needs repair or its rate needs to be weakened to log log q/log q.","tokens_in":27912,"tokens_out":34489,"duration_ms":343981,"concrete_test":"Recompute the Borda sum S(T) in the proof of Theorem 4.11 using the stated bound |bμ_i(λ)| ≪ ||λ||^{c||λ||}/|k_i|^{1/2}: bound S(T) ≤ Σ_{1≤||λ||≤T} ||λ||^{2c||λ||}/κ(λ) ≤ C T^{2cT+r} (up to constants depending on K), then optimize 1/T + |k_i|^{-1/2} T^{cT+r/2}. If this optimum is not O(1/log|k_i|), consult [35, Prop. 6.36] to determine whether the true complexity bound is c^{||λ||} or ||λ||^{c||λ||}; if the latter, Theorem 4.11's proof requires a different argument and the stated logarithmic rate is unproven.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 4.3, proof of Theorem 4.11: after quoting [26, Th. 28.1] and [35, Prop. 6.36], the authors assert |bμ_i(λ)| ≪ ||λ||^{c||λ||}/|k_i|^{1/2}. The very next display bounds the Borda sum by (Σ ||λ||^2/κ(λ) c^{2||λ||})^{1/2}; this would be valid only for a bound of the form c^{||λ||}. From ||λ||^{c||λ||} with ||λ|| ≤ T one can only infer ||λ||^{c||λ||} ≤ T^{cT}, so the sum is at most T^{O(T)} (after the O(T^r) lattice-point count), and its square root is T^{O(T)}. Inserting this into W1 ≤ 1/T + |k_i|^{-1/2} T^{O(T)} and optimizing does not give the stated O(1/log|k_i|): with T=α log|k_i| the error term is q^{O(α log log q)-1/2}, which fails for large q, while T~log q/log log q gives only O(log log q/log q). The final remark confirms the exponential-in-λ complexity is the intended mechanism; hence the displayed bound, not the summation, needs to be corrected or the theorem's rate weakened. The same gap affects the trace-pushforward inequality (19).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Wasserstein distances as an intrinsic, coordinate-free quantitative measure of equidistribution, complementing classical discrepancy bounds. After reviewing basic metric properties and proving a Bobkov-Ledoux type inequality in the torus (Appendix), the authors give two main applications. Section 3 treats empirical measures of ultra-short exponential sums attached to a finite set of algebraic integers, proving W_1(ν_p, μ_Z) ≪_Z |p|^{-1/[K_Z:Q]} (Theorem 3.1, Corollary 3.6) and, for growing prime-order subgroups of F_q^×, a Gaussian limit under d = o(log q / log log q) (Theorem 3.8). Section 4 applies Borda's inequality for compact connected Lie groups to Deligne's equidistribution theorem for curves, obtaining W_1(μ_i, μ_K) ≪ |k_i|^{-1/dim(K)} (Theorem 4.4), and to Katz's Mellin-transform equidistribution theorem, claiming a logarithmic rate (Theorem 4.11). The paper closes with a shrinking-target application for hyper-Kloosterman sums and a self-contained appendix.","tokens_in":28212,"tokens_out":20195,"duration_ms":200435,"significance":"If the results hold, the paper makes a convincing case that Wasserstein metrics are a useful quantitative equidistribution tool in analytic number theory. The strengths are substantial: the constants in Section 3 (C_Z, C'_Z) are explicit and parameter-free; the rates in Theorems 3.1 and 4.4 are concrete and falsifiable; Section 3.5 gives a nontrivial application to a regime where both q and the subgroup order d vary; and the appendix provides a complete proof of the Bobkov-Ledoux inequality. The proof of Theorem 4.4 is clean and correctly transfers Riemann Hypothesis bounds into a W_1 rate. However, the proof of Theorem 4.11 contains a genuine gap: a super-exponential Fourier bound is incorrectly inserted into Borda's inequality as an exponential bound, so the stated 1/log|k_i| rate is not established. In addition, the statement of Theorem 3.1 omits a hypothesis that is needed for its truth. Both issues are local and fixable, but they affect advertised central claims.","major_comments":[{"comment":"The Fourier bound obtained from [26, Th. 28.1] and [35, Prop. 6.36] is |b̂μ_i(λ)| ≪ ||λ||^{c||λ||}/|k_i|^{1/2}. In the next display this is inserted into Borda's inequality as (Σ_{1≤||λ||≤T} (||λ||^2/κ(λ)) c^{2||λ||})^{1/2}, i.e. the factor ||λ||^{c||λ||} is replaced by c^{||λ||}. That replacement is invalid: for ||λ||≤T one can only bound ||λ||^{c||λ||} by T^{cT}, which is not O(c^T). Correctly summing gives an error term |k_i|^{-1/2} T^{O(T)}. Taking T=α log|k_i| yields |k_i|^{-1/2} (log|k_i|)^{O(α log log|k_i|)}, which is not o(1/log|k_i|) because the exponent becomes positive when α log log|k_i| exceeds 1/2; taking T~log|k_i|/log log|k_i| gives only O(log log|k_i|/log|k_i|). Therefore the claimed O(1/log|k_i|) rate in Theorem 4.11 and in the trace-pushforward inequality (19) is not justified by the argument as written. The Fourier coefficient bound must be strengthened to a genuine exponential-in-||λ|| bound, or the theorem's rate must be weakened.","section":"Section 3.1, statement of Theorem 3.1"},{"comment":"Theorem 3.1 is stated for all prime ideals of residual degree 1, but the proof, via Corollary 3.6, requires the additional condition p∈S_Z, i.e. that no two distinct elements of Z are congruent modulo p. The statement is false without this condition: for Z={0,2} and p=(2) in O_Z=Z, the measure ν_p is the average of δ_0 and δ_2, whereas μ_Z is the uniform measure on the circle centered at 1 of radius 1 (the relation module R_Z forces f(0)=1 and leaves f(2) free). The W_1 distance between these two measures is a positive constant, whereas |p|^{-1/[K_Z:Q]} = 1/2. The theorem should be restated with the hypothesis p∈S_Z (or 'for all but finitely many p').","section":"Section 3.1, statement of Theorem 3.1"}],"minor_comments":[{"comment":"In the proof of Lemma 3.3, the expression 'bλg(α)' appears in the first sentence after the proof begins; it should be 'bλZ(α)'.","section":"Lemma 3.3"},{"comment":"In the definition of R(χ;p), the set 'F×p {1}' should read 'F×p \\ {1}'.","section":"Example 4.12"},{"comment":"When the Lambert W function is invoked, the sentence identifies W_0 only as 'the inverse bijection to x↦xe^x on [-1/e,+∞)'; it would be clearer to state explicitly that W_0 denotes the principal branch, since the equation x e^x = y has two real solutions for y∈(-1/e,0).","section":"Proof of Theorem 3.8"},{"comment":"The reference for T. Bonis (listed as [5]) lacks page numbers and possibly a DOI; please complete the bibliographic data if available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about Theorem 4.11 is legitimate and should be conveyed firmly to the authors. The rest of the paper appears sound: I checked the Section 3 arguments, the Borda inequality application in Theorem 4.4, and the appendix, and found no comparable gap. The statement error in Theorem 3.1 is embarrassing but trivial to repair. Given the otherwise high quality, I recommend major revision rather than rejection, with the expectation that the authors can either prove a genuine exponential Fourier bound in the Mellin-transform setting or restate Theorem 4.11 with the weaker rate that the current method actually supports."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my take on Kowalski–Untrau. The paper is worth reading and eventually publishing, but don't trust Theorem 4.11 as written. The proof has an internal inconsistency that the reader's report missed. After quoting [26] and [35], they claim |bμ_i(λ)| ≪ ||λ||^{c||λ||}/|k_i|^{1/2}. In the very next display they bound the Borda sum using c^{2||λ||}, which would be legitimate only for a Fourier bound of c^{||λ||}. From ||λ||^{c||λ||} with ||λ|| ≤ T you only get T^{cT}, so the sum is T^{O(T)}, and optimizing gives no 1/log q rate. The trace-pushforward version (19) has the same problem. This is a load-bearing flaw for Section 4.3, not a typo.\n\nThat said, the rest of the paper is genuinely good. Theorem 3.1 upgrades their earlier qualitative work to an explicit |p|^{-1/[K:Q]}, the proof is clean, and the constant C_Z is explicit. The central-limit regime in Theorem 3.8 is new and the d = o(log q/log log q) window is handled carefully. Theorems 4.4 and 4.9 are solid applications of Borda's inequality; the rates |k_i|^{-1/dim K} follow from the lattice-point count, and the authors are honest about the connectedness restriction in Remark 4.2(1). I checked the Fourier-vanishing step and the choice of T in Proposition 3.4; it works.\n\nThe soft spots are the one in 4.11 and the usual caveats about monodromy hypotheses. Nothing else smelled circular or fitted. The citation pattern is fine, and the paper credits prior work properly.\n\nWho is this for? Number theorists working on equidistribution and exponential sums, and anyone interested in Wasserstein methods in analytic number theory. It deserves a serious referee: the good parts are strong enough that the gap in 4.11 shouldn't sink the whole paper, but the authors need to either correct the Fourier bound (maybe the actual bound from [26, Th. 28.1] is exponential in ||λ||, which would save the argument) or weaken the theorem's rate. If the bound is truly ||λ||^{c||λ||}, the 1/log q claim is not justified.\n\nRecommendation: send to peer review, ask for a fix to 4.11 and a re-check of (19). The rest is publishable.","headline":"A strong paper with a real, localized gap in the proof of Theorem 4.11 that the authors need to fix; the main rates in Sections 3 and 4.2 look solid.","tokens_in":28891,"tokens_out":3310,"would_cite":true,"duration_ms":29842,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11K38","11L03","11T23","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Using Wasserstein metrics, the paper proves explicit polynomial equidistribution rates for ultra-short exponential sums and for Frobenius classes of lisse sheaves over finite fields.","keywords":["Wasserstein metric","quantitative equidistribution","exponential sums","trace functions","finite fields","Kloosterman sums","monodromy groups","central limit theorem"],"falsifier":"Compute, for hyper-Kloosterman sums of rank $r=2$ over the fields $\\mathbb{F}_{q^n}$ with $q$ fixed, the empirical Wasserstein distance from the uniform measure on the values $Kl_2(a;\\mathbb{F}_{q^n})$ to the semicircle law $\\frac{1}{\\pi}\\sqrt{1-x^2/4}\\,dx$ on $[-2,2]$; if this distance does not decay like $|q^n|^{-1/3}$ for infinitely many $n$, Theorem 4.4's rate for sheaves is false.","tokens_in":27623,"feed_emoji":"📏","tokens_out":18028,"duration_ms":167374,"temperature":0.7,"pith_summary":"This paper makes the case that the 1-Wasserstein metric is a natural intrinsic measuring stick for quantitative equidistribution of exponential sums over finite fields, and it proves explicit rates in two flagship settings. For a fixed finite set $Z$ of algebraic integers, the empirical measures $\\nu_p$ of the ultra-short sums $S_Z(p,a)$ converge to an explicit limit $\\mu_Z$, with $W_1(\\nu_p,\\mu_Z)\\ll_Z |p|^{-1/[K_Z:\\mathbb{Q}]}$. For lisse weight-$0$ sheaves on curves with equal, connected, bounded-complexity monodromy, the Frobenius-class measures converge to Haar measure with $W_1(\\mu_i,\\mu_K)\\ll |k_i|^{-1/\\dim(K)}$. A further theorem shows that when the subgroup size $d$ grows as $d=o(\\log q/\\log\\log q)$, normalized sums over $d$-th roots of unity tend to a standard complex Gaussian, interpolating between fixed-$d$ equidistribution and the Gaussian limit. These rates provide quantitative equidistribution in a coordinate-free form, with consequences such as shrinking-target lower bounds for hyper-Kloosterman sums.","feed_headline":"Ultra-short sums and Frobenius classes converge at explicit rates","feed_subtitle":"A coordinate-free measure turns qualitative equidistribution into explicit polynomial rates.","key_machinery":"The central object is the 1-Wasserstein metric $W_1(\\mu,\\nu)=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int d(x,y)\\,d\\pi$, whose Kantorovich–Rubinstein duality $W_1(\\mu,\\nu)=\\sup_{u\\in\\mathrm{Lip}_1}|\\int u\\,d\\mu-\\int u\\,d\\nu|$ turns equidistribution into bounds on Fourier coefficients. The mechanism is a Fourier-transport comparison: on tori, the inequality $W_1(\\mu,\\nu)\\ll \\sqrt{d}T^{-1}+(\\sum_{0<|h|_\\infty\\le T}|h|^{-2}|\\hat\\mu(h)-\\hat\\nu(h)|^2)^{1/2}$ of [3] is used, while on connected compact Lie groups the analogue of [6] gives $W_1(\\mu,\\nu)\\ll_K T^{-1}+(\\sum_{1\\le\\|\\lambda\\|\\le T}\\kappa(\\lambda)^{-1}|\\hat\\mu(\\lambda)-\\hat\\nu(\\lambda)|^2)^{1/2}$ for conjugacy-invariant measures, with $\\kappa(\\lambda)$ the Casimir eigenvalue. In Section 3, cancellation comes from a norm computation: if $\\eta_\\alpha$ is nontrivial on the relation group $H_Z$ and $|p|>C_Z\\|\\alpha\\|_1^{[K_Z:\\mathbb{Q}]}$, then the Fourier coefficient of the empirical measure vanishes. In Section 4, it comes from the Riemann Hypothesis over finite fields, which bounds $|\\hat\\mu_i(\\lambda)|$ by $|k_i|^{-1/2}$ times a Betti number controlled by the complexity bound of [35].","core_discovery":"The central discovery, stated as a theorem on the paper's own terms, is that the 1-Wasserstein metric converts two qualitative equidistribution results into quantitative ones with explicit polynomial rates. Theorem 3.1 (Corollary 3.6) shows that for a finite set $Z$ of algebraic integers, with $K_Z=\\mathbb{Q}(Z)$ and $\\mu_Z$ defined from additive relations among elements of $Z$, the empirical measures $\\nu_p$ of $S_Z(p,a)$ satisfy $W_1(\\nu_p,\\mu_Z)\\ll_Z |p|^{-1/[K_Z:\\mathbb{Q}]}$ for all prime ideals $p\\in S_Z$. Theorem 4.4 shows that for lisse weight-$0$ sheaves on curves whose arithmetic and geometric monodromy groups are equal, connected, independent of $i$, and of bounded complexity, the conjugacy-invariant Frobenius-class measures satisfy $W_1(\\mu_i,\\mu_K)\\ll |k_i|^{-1/\\dim(K)}$, and the same holds after pushing forward by the trace map. The Mellin-transform variant (Theorem 4.11) gives the logarithmic rate $W_1(\\mu_i,\\mu_K)\\ll 1/\\log|k_i|$ for families parameterized by multiplicative characters. Theorem 3.8 shows that, for prime $d\\mid q-1$ with $d\\to\\infty$ and $d=o(\\log q/\\log\\log q)$, the normalized sums $\\frac{1}{\\sqrt{d}}\\sum_{x\\in\\mu_d(\\mathbb{F}_q)}e(ax/q)$ become equidistributed as $\\mathcal{N}(0,\\frac{1}{2}I_2)$ in the complex plane; the same Gaussian conclusion is proved for $d=r^b$ with fixed prime $r$.","pith_inferences":["If the compact-Lie-group Fourier inequality of [6] is extended to disconnected compact groups, the same proof should give polynomial rates for sheaves whose monodromy is disconnected, with the exponent governed by the identity component.","The logarithmic rate in the Mellin-transform theorem (Theorem 4.11) is likely an artifact of exponential Betti-number bounds; if those bounds can be replaced by polynomial ones, the argument would yield polynomial rates and match Theorem 4.4.","The Wasserstein formulation suggests a concrete computational test: for any family of exponential sums with known monodromy group, the empirical $W_1$ distance to the expected limit should scale like $|k|^{-1/\\dim K}$, so deviations could signal a failure of the equal/connected monodromy hypothesis.","The optimal-transport interpretation raises an arithmetic question the paper leaves open: whether the optimal coupling between $\\nu_p$ and $\\mu_Z$ has a description in terms of pairs of reductions modulo primes or in terms of algebraic relations in $Z$."],"forward_implications":["For fixed $Z$, the empirical measures of ultra-short sums reach their limit $\\mu_Z$ at rate $|p|^{-1/[K_Z:\\mathbb{Q}]}$, with constants depending only on $Z$; this refines the earlier qualitative convergence.","In the vertical direction, for a fixed sheaf over $\\mathbb{F}_q$ with equal connected monodromy, the Frobenius-class measures over extensions $\\mathbb{F}_{q^n}$ converge to Haar measure at rate $|q^n|^{-1/\\dim K}$, and the pushforward by the trace gives the same rate for the discrete measures of exponential-sum values.","In the horizontal direction, as $p\\to\\infty$ through primes with bounded-complexity sheaves of equal connected monodromy, the same $p^{-1/\\dim K}$ rate holds for the corresponding exponential-sum values.","For $r=2$ Kloosterman sums the trace map identifies the limit with the semicircle measure $\\frac{1}{\\pi}\\sqrt{1-x^2/4}\\,dx$ on $[-2,2]$, so the Wasserstein distance to that measure is $O(|k_n|^{-1/3})$.","The Gaussian theorem gives a quantitative interpolation: for prime $d=o(\\log q/\\log\\log q)$, the normalized sums over $d$-th roots of unity are asymptotically standard complex normal, with the growth condition separating the obtainable regime from the regime where nontrivial bounds are impossible."],"supporting_citations":[{"why":"The authors' earlier study of ultra-short sums, which supplies the definitions of $S_Z(p,a)$, $H_Z$, and the qualitative equidistribution that Theorem 3.1 refines.","marker":"[30]"},{"why":"Supplies the Fourier inequality for the Wasserstein metric on connected compact Lie groups that carries the proof of Theorem 4.4.","marker":"[6]"},{"why":"Supplies the torus Fourier-analytic transport inequality used in Section 3 and proved in the appendix.","marker":"[3]"},{"why":"The Riemann Hypothesis over finite fields for sheaves, which gives the polynomial decay of Fourier coefficients of Frobenius-class measures.","marker":"[13]"},{"why":"Quantitative sheaf theory that defines bounded complexity and controls the sums of Betti numbers used in the rate of Theorem 4.4.","marker":"[35]"},{"why":"Establishes the qualitative equidistribution for finite-field Mellin transforms that Theorem 4.11 quantifies.","marker":"[26]"},{"why":"Provides the fixed-$d$ equidistribution of Gaussian-period-type sums that the growing-$d$ Gaussian theorem extends.","marker":"[15]"},{"why":"Shows that no nontrivial bound is possible when $d\\ll\\log q$, delimiting the growth window in Theorem 3.8.","marker":"[28]"},{"why":"Supplies the Betti-number bound used in the vertical version of the sheaf equidistribution theorem (Theorem 4.9).","marker":"[27]"}],"fun_headline_variants":["Explicit polynomial rates for exponential sum equidistribution","Wasserstein metric turns qualitative into quantitative equidistribution","Ultra-short sums and Frobenius classes converge with explicit rates","Gaussian law for normalized sums over finite field subgroups","Coordinate-free measure yields explicit decay for equidistribution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the groups involved are connected: for the sheaf equidistribution theorems, the arithmetic and geometric monodromy groups are assumed equal, connected, and independent of $i$, because the Fourier inequality used there requires connectedness, and for the Gaussian theorem the subgroup size must stay inside $d=o(\\log q/\\log\\log q)$; if either fails, the stated rate is not established.","fun_headline_variants_meta":{"raw":{"variants":["Explicit polynomial rates for exponential sum equidistribution","Wasserstein metric turns qualitative into quantitative equidistribution","Ultra-short sums and Frobenius classes converge with explicit rates","Gaussian law for normalized sums over finite field subgroups","Coordinate-free measure yields explicit decay for equidistribution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000223,"raw_usage":{"total_tokens":1484,"prompt_tokens":999,"completion_tokens":485,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":407}},"tokens_in":615,"tokens_out":485,"duration_ms":5466,"temperature":1.0,"reasoning_tokens":407,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:19:17.008700+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for hyper-Kloosterman sums of rank $r=2$ over the fields $\\mathbb{F}_{q^n}$ with $q$ fixed, the empirical Wasserstein distance from the uniform measure on the values $Kl_2(a;\\mathbb{F}_{q^n})$ to the semicircle law $\\frac{1}{\\pi}\\sqrt{1-x^2/4}\\,dx$ on $[-2,2]$; if this distance does not decay like $|q^n|^{-1/3}$ for infinitely many $n$, Theorem 4.4's rate for sheaves is false.","supporting_citations":[{"cited_title":"Kowalski and T","cited_arxiv_id":null,"evidence_quote":"The authors' earlier study of ultra-short sums, which supplies the definitions of $S_Z(p,a)$, $H_Z$, and the qualitative equidistribution that Theorem 3.1 refines."},{"cited_title":"Borda, Berry-Esseen smoothing inequality for the Wasserstein metric on compact Lie groups , J","cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier inequality for the Wasserstein metric on connected compact Lie groups that carries the proof of Theorem 4.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the torus Fourier-analytic transport inequality used in Section 3 and proved in the appendix."},{"cited_title":"Deligne, La conjecture de Weil","cited_arxiv_id":null,"evidence_quote":"The Riemann Hypothesis over finite fields for sheaves, which gives the polynomial decay of Fourier coefficients of Frobenius-class measures."},{"cited_title":"Sawin, A","cited_arxiv_id":null,"evidence_quote":"Quantitative sheaf theory that defines bounded complexity and controls the sums of Betti numbers used in the rate of Theorem 4.4."},{"cited_title":"Sato–Tate theorems for finite-field Mellin transforms , Annals of Mathematics Studies, vol","cited_arxiv_id":null,"evidence_quote":"Establishes the qualitative equidistribution for finite-field Mellin transforms that Theorem 4.11 quantifies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the fixed-$d$ equidistribution of Gaussian-period-type sums that the growing-$d$ Gaussian theorem extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that no nontrivial bound is possible when $d\\ll\\log q$, delimiting the growth window in Theorem 3.8."},{"cited_title":"Katz and P","cited_arxiv_id":null,"evidence_quote":"Supplies the Betti-number bound used in the vertical version of the sheaf equidistribution theorem (Theorem 4.9)."}],"review_version":1}