Distribution of solutions to systems of congruences in balls

Michael Harm

arxiv: 2403.05078 · v2 · pith:W5FAPKSEnew · submitted 2024-03-08 · 🧮 math.NT

Distribution of solutions to systems of congruences in balls

Michael Harm This is my paper

Pith reviewed 2026-05-24 02:48 UTC · model grok-4.3

classification 🧮 math.NT

keywords equidistributionWeyl sumsfinite fieldsdiscrepancyvariancepolynomialscongruencestorus

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The pith

Under non-trivial Weyl sum bounds, the fractional parts of polynomial evaluations over large finite fields show refined equidistribution with explicit ball discrepancy and variance bounds.

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

The paper studies the distribution of points in the n-dimensional torus given by the fractional parts of n polynomials evaluated at all points in a finite field with p elements, where p is a large prime. Assuming non-trivial estimates for the associated Weyl sums, it establishes bounds on the discrepancy of these points when counted inside balls and on the variance of such counts. This refines standard equidistribution results for the solutions to systems of polynomial congruences. A sympathetic reader would care because these bounds give quantitative control over how uniformly the solutions are spread in geometric regions, which is useful for counting problems in number theory.

Core claim

For sufficiently large primes p and given non-trivial bounds on the Weyl sums for non-trivial linear combinations of the polynomials G1 to Gn in m variables over F_p, the vectors of fractional parts ({G1(x)/p}, ..., {Gn(x)/p}) for x in F_p^m exhibit refined equidistribution properties, including bounds for the ball discrepancy and the variance.

What carries the argument

Non-trivial bounds on Weyl sums for non-trivial linear combinations of the G polynomials, which are used to derive the discrepancy and variance estimates for the distribution in the torus.

If this is right

The ball discrepancy of the point set in the torus is bounded in terms of the Weyl sum bounds.
The variance of the number of points in balls is controlled similarly.
Equidistribution refinements hold for the image of the polynomial map in the torus.
These properties apply to the distribution of solutions to the system of congruences defined by the polynomials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the Weyl sum bounds can be verified for specific families of polynomials, the distribution results would apply directly to those cases.
Such distribution bounds might extend to counting lattice points or estimating exponential sums in related arithmetic settings.
The results suggest that stronger Weyl estimates could lead to sharper discrepancy bounds.

Load-bearing premise

Non-trivial bounds exist for the Weyl sums associated to the non-trivial linear combinations of the polynomials.

What would settle it

An explicit counterexample where, for some polynomials satisfying the Weyl sum assumptions for large p, the ball discrepancy or variance exceeds the derived bounds.

read the original abstract

Let $G_1,\dots, G_n\in \mathbb{F}_p[X_1,\dots,X_m]$ be $n$ polynomials in $m$ variables over the finite field $\mathbb{F}_p$ of $p$ elements. For any sufficiently large prime $p$ and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of $G=(G_1,\dots, G_n)$, we study various properties regarding the distribution of the vectors by fractional parts \begin{equation*} \bigg(\bigg\{ \frac{G_1(\textbf{x})}{p}\bigg\},\cdots,\bigg\{ \frac{G_n(\textbf{x})}{p}\bigg\}\bigg)\in \mathbb{T}^n,\hspace{10pt} \textbf{x}\in \mathbb{F}_p^m. \end{equation*} We prove refinements of equidistribution, such as bounds for the ball discrepancy and variance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper derives conditional ball-discrepancy and variance bounds for polynomial images over F_p from assumed Weyl-sum estimates, but adds only incremental refinements to existing equidistribution statements.

read the letter

The main point is that the results on distribution of the vectors ({G1(x)/p}, ..., {Gn(x)/p}) inside balls are conditional on non-trivial bounds for the Weyl sums attached to nontrivial linear combinations of the Gi. Under that hypothesis the paper produces explicit discrepancy and variance estimates in the usual way for finite-field equidistribution problems. That is the actual content. The setup with n polynomials in m variables is handled uniformly, and the translation from sum bounds to geometric discrepancy statements is carried out without obvious gaps. The proofs are presented as standard applications of the assumed estimates rather than new machinery. The limitation is that the work supplies no new information on the Weyl sums themselves, so the discrepancy bounds remain unavailable until those estimates are obtained elsewhere. The abstract frames the contribution explicitly as refinements, which matches what is delivered. No internal contradictions appear in the conditional argument, and the dependence on the external hypothesis is stated up front. This is for readers already working on exponential sums and distribution problems over finite fields who need explicit constants or variance statements under known sum bounds. It does not open new directions or remove the main obstruction in the area. A serious referee should see it, since the derivation is clean and the scope is clearly delimited; the conditional nature lets reviewers judge the incremental value directly.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that, assuming non-trivial bounds on the Weyl sums associated to non-trivial linear combinations of given polynomials G = (G1, ..., Gn) in m variables over F_p, for sufficiently large primes p the vectors ({G1(x)/p}, ..., {Gn(x)/p}) for x in F_p^m exhibit refined equidistribution in the n-torus, including explicit bounds on ball discrepancy and variance.

Significance. Conditional on the Weyl-sum hypotheses, the work supplies standard but useful refinements (discrepancy in balls, variance) to equidistribution statements for polynomial maps over finite fields; such conditional results are of interest in analytic number theory when the underlying exponential-sum bounds can be verified in concrete cases.

minor comments (2)

[Abstract] Abstract: the phrase 'non-trivial bounds for the Weyl sums' is left informal; a brief indication of the expected saving (e.g., |sum| ≪ p^{m-δ} for δ>0) would clarify the strength of the hypothesis without lengthening the statement.
[Introduction] The transition from the Weyl-sum hypothesis to the ball-discrepancy bound is described only at the level of the abstract; a short outline of the reduction (e.g., via the Erdős–Turán inequality or Fourier analysis on the torus) in the introduction would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were provided in the report, so we have no individual points to address point-by-point. The manuscript already contains the claimed results under the stated Weyl sum hypotheses.

Circularity Check

0 steps flagged

No significant circularity; results conditional on external Weyl-sum bounds

full rationale

The paper states its main results explicitly as conditional on the availability of non-trivial bounds for the Weyl sums associated to non-trivial linear combinations of the polynomials G. The claimed refinements (ball discrepancy and variance bounds) are then derived from those external hypotheses using standard equidistribution techniques over finite fields. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via prior work appear in the provided abstract or description. The derivation chain therefore remains self-contained against the stated external assumptions and does not reduce its conclusions to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the external hypothesis of non-trivial Weyl-sum bounds; no free parameters, invented entities, or additional axioms are visible in the abstract.

axioms (1)

domain assumption Non-trivial bounds exist for the Weyl sums attached to non-trivial linear combinations of the given polynomials.
This hypothesis is explicitly required in the abstract to obtain the discrepancy and variance bounds.

pith-pipeline@v0.9.0 · 5681 in / 1128 out tokens · 30677 ms · 2026-05-24T02:48:58.283600+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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[1] [1]

Apostol, Introduction to Analytic Number Theory, Springer, New York (2013)

T. Apostol, Introduction to Analytic Number Theory, Springer, New York (2013)

work page 2013

[2] [2]

Cohn, Measure Theory, Springer, New York (2013)

D. Cohn, Measure Theory, Springer, New York (2013)

work page 2013

[3] [3]

Deligne, La conjecture de Weil: I, Publ

P. Deligne, La conjecture de Weil: I, Publ. Math. IH\'ES, 43 (1974), 273--307

work page 1974

[4] [4]

Fouvry and N

\'E. Fouvry and N. Katz, A general stratification theorem for exponential sums, and applications, J. Reine Angew. Math., 540 (2001), 115--166

work page 2001

[5] [5]

Grafakos, Classical Fourier Analysis, Springer, New York (2014)

L. Grafakos, Classical Fourier Analysis, Springer, New York (2014)

work page 2014

[6] [6]

Humphries, Distributing points on the torus via modular inverses, Quart

P. Humphries, Distributing points on the torus via modular inverses, Quart. J. Math., 73 (2022), 1--16

work page 2022

[7] [7]

Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., Vol. 53, Providence (2004)

work page 2004

[8] [8]

Kerr and I

B. Kerr and I. E. Shparlinski, On the distribution of values and zeros of polynomial systems over arbitrary sets, J. Number Theory, 133 (2013), 2863--2873

work page 2013

[9] [9]

B. G. Korenev, Bessel Functions and Their Applications, CRC Press (2002)

work page 2002

[10] [10]

Kuipers and H

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, New York (1974)

work page 1974

[11] [11]

W.-C. W. Li, Number Theory with Applications, World Scientific, Singapore (1996)

work page 1996

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Makarov and A

B. Makarov and A. Podkorytov, Real Analysis: Measures, Integrals and Applications, Springer, London (2013)

work page 2013

[13] [13]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, PMS-32, Princeton University Press, Princeton (1972)

work page 1972

[14] [14]

Wen-Ching Winnie Li, Number theory with applications, World Scientific, Singapore, 1996

work page 1996

[15] [15]

Michael Jorgensen, Volumes of n-dimensional spheres and ellipsoids (2014)

work page 2014

[16] [16]

Zdravko Cvetkovski, H \"o lder's Inequality, Minkowski's Inequality and Their Variants , Springer, Berlin, Heidelberg (2012)

work page 2012

[17] [17]

Math., Volume 21 (1919), 39--60

Raymond Brink, A New Sequence of Integral Tests for the Convergence and Divergence of Infinite Series, Ann. Math., Volume 21 (1919), 39--60

work page 1919

[18] [18]

George Watson, A treatise on the theory of Bessel functions, The University Press, Volume 2 (1922)

work page 1922

[19] [19]

Edmund Whittaker, George Watson, A course of modern analysis: Third Edition, Cambridge University Press, Cambridge (1920)

work page 1920