Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Wasserstein metrics and quantitative equidistribution of exponential sums over finite fields

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.22059 v2 pith:VUVVGDNB submitted 2025-05-28 math.NT

Emmanuel Kowalski , Théo Untrau This is my paper
classification math.NT MSC 11K3811L0311T2349Q22
keywords WassersteinmetricquantitativeequidistributionexponentialsumstracefunctionsfinitefieldsKloostermanmonodromygroupscentrallimittheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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].

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $r$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $Z$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Section 3.1, statement of Theorem 3.1] 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.
  2. [Section 3.1, statement of Theorem 3.1] 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').
minor comments (4)
  1. [Lemma 3.3] 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)'.
  2. [Example 4.12] In the definition of R(χ;p), the set 'F×p {1}' should read 'F×p \ {1}'.
  3. [Proof of Theorem 3.8] 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).
  4. [References] The reference for T. Bonis (listed as [5]) lacks page numbers and possibly a DOI; please complete the bibliographic data if available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quantitative rates are derived from external spectral estimates and explicit, non-fitted constants; self-citations are independent prior results.

full rationale

We find no circular derivation in the paper. In Section 3, the limiting measure μ_Z is defined directly from the additive-relation group H_Z, and the Fourier coefficients of λ_Z are computed from the definition of Haar measure (Lemma 3.2); the empirical coefficients bμ_p(α) are evaluated by orthogonality of characters (Lemma 3.3), with the explicit constant C_Z = ∏_σ max_{x∈Z}|σ(x)| bounding the norm of γ(α). No parameter is fitted: T is chosen as (|p|/C_Z)^{1/[K_Z:Q]}, and the rate |p|^{-1/[K_Z:Q]} follows. The pushforward to C uses the explicit Lipschitz constant √|Z|. Section 3.5 invokes [30, §3] only to identify R_{μ_d} for prime d; that is an external, published algebraic fact and does not assume the quantitative Wasserstein estimate being proved. In Section 4, Borda's inequality (Theorem 4.1) is quoted from an external source and then combined with Deligne's Riemann Hypothesis and Katz's bounds to bound Fourier coefficients; the Haar measure on K is the benchmark, not a fitted object. The constant T = |k_i|^{1/n} is chosen, not fitted. The apparent issue flagged in Theorem 4.11's proof (the replacement of ||λ||^{c||λ||} by c^{||λ||} in the Borda sum) is a possible error in the summation step, not a circularity: the bound is imported from [26] and [35], and the conclusion does not reduce to an input by definition. Hence a score of 0 is appropriate.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

Every constant in the main theorems is explicit (e.g. C_Z = Π_σ max_{x∈Z}|σ(x)| and C'_Z = 4√3√|Z|(|Z|+1)C_Z^{1/[K:Q]}), and the several auxiliary scales T are chosen, not fitted. The targets are external benchmarks: Haar measure on H_Z (Fourier coefficients computed from its definition), Haar measure on K, and the complex Gaussian from the CLT. No axiom assumes the target result, and the self-cited results ([30, §3]) are used only for the structure of the additive relation module.

assumptions (6)
  • domain assumption Deligne's Riemann Hypothesis over finite fields (Weil II), used in the form of the Fourier coefficient bound (17) in the proof of Theorem 4.4.
    Quoted from Katz [25, Ch. 3] and Katz-Sarnak [27, Th. 9.2.6]; it converts sums of trace functions into Betti-number-controlled estimates and is the engine behind the rates in Section 4. It is an external theorem, not reproved in the paper.
  • domain assumption Borda's Fourier-Wasserstein inequality for connected compact Lie groups (Theorem 4.1, from [6, Th. 1]).
    The entire Section 4 uses this inequality as its starting point; the restriction to connected K is explicitly acknowledged in Remark 4.2(1).
  • domain assumption Betti number and complexity bounds: σ_c(X_i, ϱλ(F_i)) ≪ dim(ϱλ) c(F_i) ([35, Rem. 6.34]) and the Mellin-transform complexity bound ([35, Prop. 6.36]).
    These bounds turn the Riemann Hypothesis estimate into the sum over λ that yields the |ki|^{-1/n} rate in Theorem 4.4 and the 1/log|ki| rate in Theorem 4.11; cited from the published quantitative sheaf theory paper [35].
  • domain assumption Katz's monodromy computations, e.g. [25, Th. 11.1] for hyper-Kloosterman sheaves and [26, Th. 14.5] for the Mellin sheaf with trace R(χ;p).
    Used only in the examples to identify the compact group K as SUr(C) or USpr(C); the main theorems are stated for arbitrary K satisfying their hypotheses.
  • domain assumption For prime d, the additive relation module R_µd of the d-th roots of unity is generated by the constant map ([30, §3]).
    Used in Lemma 3.9 to identify H_µd with (S^1)^{d-1}, so the limiting measure γ_d is the pushforward of Haar measure by the Laurent polynomial g_d; the fact is taken from the authors' prior paper by reference.
  • standard math Kantorovich-Rubinstein duality and the metrization of weak convergence by Wasserstein distances (Villani [40, Th. 1.14, Th. 7.12]).
    These are standard optimal transport background used in Theorem 1.2 (properties (1) and (6)) and in Lemma 3.10 for the Wasserstein central limit theorem.
invented entities (1)
  • The compact group H_Z = {f: Z → S^1 : α ∈ R_Z ⇒ Π_x f(x)^{α(x)} = 1} and its pushforward µ_Z = σ_*λ_Z under summation. independent evidence
    purpose: Defines the limiting distribution for the empirical measures ν_p of ultra-short sums; the paper proves W_1(ν_p, µ_Z) ≪_Z |p|^{-1/[K:Q]}.
    Not a hat-pulled object: H_Z is determined by the additive relations R_Z among the given algebraic integers, and its Haar measure has explicit Fourier coefficients that provably match the empirical Fourier coefficients of ν_p in the limit (Lemmas 3.2 and 3.3). The limit measure is computed concretely in examples, e.g. (S^1)^{d-1} pushed forward by the Laurent polynomial g_d.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Wasserstein metrics and quantitative equidistribution of exponential sums over finite fields." pith.science (2026). https://pith.science/paper/VUVVGDNB

@misc{pith2026250522059,
  author       = {Pith},
  title        = {Pith review of: Wasserstein metrics and quantitative equidistribution of exponential sums over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VUVVGDNB}},
  note         = {Machine review of arXiv:2505.22059}
}
read the original abstract

The Wasserstein distance between probability measures on compact spaces provides a natural invariant quantitative measure of equidistribution, which is partly similar to the classical discrepancy appearing in Erd\"os-Tur\'an type inequalities in the case of tori, but is a more intrinsic quantity. We recall the basic properties of Wasserstein distances and present applications to quantitative forms of equidistribution of exponential sums in two examples, one related to our previous work on the equidistribution of ultra-short exponential sums, and the second a quantitative form of the equidistribution theorems of Deligne and Katz.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The asymptotic Mahler measure of Gaussian periods

    math.NT 2025-07 accept novelty 8.0 of 10

    For fixed k, the asymptotic Mahler measure of Gaussian periods of conductor kn+1 is n times the Mahler measure of the cyclovariety x0+F_k(x)=0, which is asymptotically (1/2)log k.

Reference graph

Works this paper leans on

42 extracted references · 41 canonical work pages · cited by 1 Pith paper

  1. [26]

    Sato–Tate theorems for finite-field Mellin transforms , Annals of Mathematics Studies, vol

    , Convolution and equidistribution. Sato–Tate theorems for finite-field Mellin transforms , Annals of Mathematics Studies, vol. 180, Princeton University Press, Princeton, NJ, 2012

    work page 2012

  2. [35]

    Sawin, A

    W. Sawin, A. Forey, J. Fres´ an, and E. Kowalski, Quantitative sheaf theory , J. Amer. Math. Soc. 36 (2023), no. 3, 653–726

    work page 2023

  3. [1]

    Billingsley, Probability and measure, Third, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995

    P. Billingsley, Probability and measure, Third, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication

    work page 1995

  4. [2]

    S. G. Bobkov and M. Ledoux, Transport inequalities on Euclidean spaces for non-Euclidean metrics , J. Fourier Anal. Appl. 26 (2020), no. 4, 27

    work page 2020

  5. [3]

    , A simple Fourier analytic proof of the AKT optimal matching theorem , Ann. Appl. Probab. 31 (2021), no. 6, 2567–2584

    work page 2021

  6. [4]

    Bonis, Stein ’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem , Probab

    T. Bonis, Stein ’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem , Probab. Theory Relat. Fields 178 (2020), no. 3-4, 827–860

    work page 2020

  7. [5]

    , Improved rates of convergence for the multivariate central limit theorem in Wasserstein distance, Electron. J. Probab. 29 (2024)

    work page 2024

  8. [6]

    Borda, Berry-Esseen smoothing inequality for the Wasserstein metric on compact Lie groups , J

    B. Borda, Berry-Esseen smoothing inequality for the Wasserstein metric on compact Lie groups , J. Fourier Anal. Appl. 27 (2021), no. 2, 24

    work page 2021

Show all 42 references
  1. [7]

    II: The Wasserstein metric , Bernoulli 27 (2021), no

    , Equidistribution of random walks on compact groups. II: The Wasserstein metric , Bernoulli 27 (2021), no. 4, 2598–2623

    2021

  2. [8]

    Bourbaki, ´El´ ements de math´ ematique

    N. Bourbaki, ´El´ ements de math´ ematique. Groupes et alg` ebres de Lie. Chapitre 9, Masson, Paris, 1982

    1982

  3. [9]

    Th´ eories spectrales

    , ´ el´ ements de math´ ematique. Th´ eories spectrales. Chapitres 3 ` a 5, Springer, 2023

    2023

  4. [10]

    Bourgain, A

    J. Bourgain, A. A. Glibichuk, and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order , J. London Math. Soc. (2) 73 (2006), no. 2, 380–398

    2006

  5. [11]

    Brown and S

    L. Brown and S. Steinerberger, On the Wasserstein distance between classical sequences and the Lebesgue measure, Trans. Am. Math. Soc. 373 (2020), 8943–8962

    2020

  6. [12]

    Burkhardt, A Z-Y

    P. Burkhardt, A Z-Y. Chan, G. Currier, S. R. Garcia, F. Luca, and H. Suh, Visual properties of gener- alized Kloosterman sums , J. Number Theory 160 (2016), 237–253. 35

    2016

  7. [13]

    Deligne, La conjecture de Weil

    P. Deligne, La conjecture de Weil. II , Inst. Hautes ´Etudes Sci. Publ. Math. 52 (1980), 137–252

    1980

  8. [14]

    Drmota and R

    M. Drmota and R. F. Tichy, Sequences, discrepancies and applications , Lect. Notes Math., vol. 1651, Berlin: Springer, 1997

    1997

  9. [15]

    W. Duke, S. R. Garcia, and B. Lutz, The graphic nature of Gaussian periods , Proc. Am. Math. Soc. 143 (2015), no. 5, 1849–1863

    2015

  10. [16]

    Erd¨ os and P

    P. Erd¨ os and P. Tur´ an,On a problem in the theory of uniform distribution. I , Nederl. Akad. Wetensch., Proc. 51 (1948), 1146–1154 = Indagationes Math. 10, 370–378

    1948

  11. [17]

    Forey, J

    A. Forey, J. Fres´ an, and E. Kowalski, Arithmetic fourier transforms over finite fields , 2021. preprint, arXiv:2109.11961

    2021 arXiv

  12. [18]

    Fouvry and Ph

    ´E. Fouvry and Ph. Michel, `A la recherche de petites sommes d’exponentielles, Ann. Inst. Fourier (Greno- ble) 52 (2002), no. 1, 47–80

    2002

  13. [19]

    , Sommes de modules de sommes d’exponentielles , Pacific J. Math. 209 (2003), no. 2, 261–288. Erratum, Pacific J. Math. 225 (2006), no. 1, 199–200

    2003

  14. [20]

    L. Fu, Y-K. Lau, and P. Xi, An effective Deligne’s equidistribution theorem (2025). preprint, arXiv: 2406.09633v2

    2025 arXiv

  15. [21]

    Ringeling G

    B. Ringeling G. Cornelissen D. Hokken, The asymptotic Mahler measure of Gaussian periods (2025). preprint, arXiv:2507.09303

    2025 arXiv

  16. [22]

    S. R. Garcia, T. Hyde, and B. Lutz, Gauss’s hidden menagerie: from cyclotomy to supercharacters , Notices Am. Math. Soc. 62 (2015), no. 8, 878–888

    2015

  17. [23]

    Graham, Irregularity of distribution in Wasserstein distance , J

    C. Graham, Irregularity of distribution in Wasserstein distance , J. Fourier Anal. Appl. 26–75 (2020)

    2020

  18. [24]

    Gray, Tubes, Second, Progress in Mathematics, vol

    A. Gray, Tubes, Second, Progress in Mathematics, vol. 221, Birkh¨ auser Verlag, Basel, 2004

    2004

  19. [25]

    Katz, Gauss sums, Kloosterman sums, and monodromy groups , Annals of Mathematics Studies, vol

    N.M. Katz, Gauss sums, Kloosterman sums, and monodromy groups , Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988

    1988

  20. [27]

    Katz and P

    N.M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy , American Math- ematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999

    1999

  21. [28]

    S. V. Konyagin, Exponential sums over multiplicative groups in fields of prime order and related com- binatorial problems , Lecture notes available at https://www.mathtube.org/sites/default/files/ lecture-notes/Konyagin_Lectures.pdf

  22. [29]

    Kowalski and W.F

    E. Kowalski and W.F. Sawin, Kloosterman paths and the shape of exponential sums , Compos. Math. 152 (2016), no. 7, 1489–1516

    2016

  23. [30]

    Kowalski and T

    E. Kowalski and T. Untrau, Ultra-short sums of trace functions , Acta Arith. 210 (2023), 367–390

    2023

  24. [31]

    Kurlberg, L

    P. Kurlberg, L. Rosenzweig, and Z. Rudnick, Matrix elements for the quantum cat map: fluctuations in short windows , Nonlinearity 20 (2007), no. 10, 2289–2304

    2007

  25. [32]

    Neukirch, Algebraic number theory

    J. Neukirch, Algebraic number theory. Transl. from the German by Norbert Schappacher , Grundlehren Math. Wiss., vol. 322, Berlin: Springer, 1999

    1999

  26. [33]

    Niederreiter, The distribution of values of Kloosterman sums , Arch

    H. Niederreiter, The distribution of values of Kloosterman sums , Arch. Math. (Basel) 56 (1991), no. 3, 270–277

    1991

  27. [34]

    Platt, Visual aspects of Gaussian periods and analogues , Int

    S. Platt, Visual aspects of Gaussian periods and analogues , Int. J. Number Theory 20 (2024), no. 9, 2227–2266

    2024

  28. [36]

    Shi and B

    Y. Shi and B. Xu, Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary, Ann. Glob. Anal. Geom. (2010), 2–26

    2010

  29. [37]

    Steinerberger, Wasserstein distance, Fourier series and applications , Monatsh

    S. Steinerberger, Wasserstein distance, Fourier series and applications , Monatsh. Math. 194 (2021), no. 2, 305–338

    2021

  30. [38]

    Untrau, Study of the distribution of some short exponential sums , Ph.D

    T. Untrau, Study of the distribution of some short exponential sums , Ph.D. thesis, Universit´ e de Bor- deaux, https://theses.hal.science/tel-04230646 (2023)

    2023

  31. [39]

    , Equidistribution of exponential sums indexed by a subgroup of fixed cardinality , Math. Proc. Camb. Philos. Soc. 176 (2024), no. 1, 65–94. 36

    2024

  32. [40]

    Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol

    C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Math- ematical Society, Providence, RI, 2003

    2003

  33. [41]

    Weyl, ¨Uber die Gleichverteilung von Zahlen mod

    H. Weyl, ¨Uber die Gleichverteilung von Zahlen mod. Eins. , Math. Ann. 77 (1916), 313–352

    1916

  34. [42]

    Xi, Equidistribution of Jacobi sums (2018)

    P. Xi, Equidistribution of Jacobi sums (2018). preprint, arXiv:1809.04286. ETH Z¨urich – D-MATH, R¨amistrasse 101, 8092 Z¨urich, Switzerland Email address : kowalski@math.ethz.ch Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France Email address : theo.untrau@ens-rennes.fr 37

    2018 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.