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Randomly piercing algebraic sets

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Sampling rac{\log p}{2 \log(1+(p-1)^{-1})} n^2 random points in F_p^n intersects every quadratic hypersurface almost surely as n grows.

desk verdict The paper derives an explicit sharp threshold for random points to hit all bounded-degree algebraic sets in F_p^n, and the error analysis looks tight enough to make the leading constant exact. read the letter →

arxiv 2606.19677 v1 pith:LPP7YROD submitted 2026-06-18 math.NT math.CO

Daniel Altman , Nathan Tung This is my paper · ORCID
classification math.NTmath.CO
keywords algebraicsetsfinitefieldsquadratichypersurfacesprobabilisticmethodthresholdphenomenaSzemeréditheoremrandomsamplingpiercing
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

The paper determines a sharp threshold on the number of random points one must sample from $F_p^n$ so that the resulting set intersects every algebraic set cut out by at most s polynomials of degree at most k. For the concrete case of a single quadratic equation the threshold is asymptotically $\frac{\log p}{2 \log(1+(p-1)^{-1})} n^2$ points; sampling $o(n^2)$ fewer points leaves some quadratic hypersurface untouched with high probability. The same probabilistic argument supplies improved explicit constants in lower bounds for the random Szemerédi theorem over these spaces. A reader cares because the result gives a precise quantitative answer to how many samples are required to guarantee that a random set meets every low-degree algebraic constraint.

What carries the argument

The probabilistic sampling model together with union-bound or deletion-method estimates on the probability that a random point set misses a fixed algebraic set of bounded degree and number of equations.

What would settle it

For a fixed small prime $p$, compute or estimate the minimal $m(n)$ such that some set of $m(n)$ points in $F_p^n$ misses at least one quadratic hypersurface, and check whether $m(n) / n^2$ tends to a limit strictly larger or smaller than $\frac{\log p}{2 \log(1+(p-1)^{-1})}$.

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Extended reading notes

Core claim

The central claim is a sharp threshold for the following problem: how many points in $F_p^n$ does one need to randomly sample to almost surely intersect every algebraic set defined by at most s polynomials each of degree at most k? In particular, sampling $\frac{\log p}{2 \log(1+(p-1)^{-1})} \cdot n^2 (1+o(1))$ points intersects every quadratic hypersurface asymptotically almost surely, while sampling $o(n^2)$ fewer points almost surely fails to intersect some quadratic hypersurface. The proof uses probabilistic deletion or union-bound arguments that track the probability a random set misses a given zero set.

Load-bearing premise

Uniform random sampling in the regime n to infinity with p fixed lets the union-bound or deletion arguments recover the exact leading constant without further error-term control.

Editorial extensions

If this is right

  • The same threshold statement holds for algebraic sets cut out by any fixed number s of polynomials of any fixed degree k.
  • The resulting lower bounds for the random Szemerédi theorem in F_p^n improve on previous work, with the explicit constant growing as the density parameter shrinks.
  • The leading constant is fully explicit and depends only on p, s and k.

Reading between the lines

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

  • The deletion-method technique could be adapted to give thresholds for hitting algebraic sets of higher codimension or for other notions of density in finite geometries.
  • Numerical checks for small p and moderate n would reveal how quickly the asymptotic regime sets in.
  • The same counting arguments might produce thresholds in related piercing problems over rings other than finite fields.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper establishes sharp thresholds for the number of random points in ℝ_p^n needed to intersect every algebraic set defined by at most s polynomials each of degree at most k. For quadratic hypersurfaces (s=1, k=2), sampling \frac{\log p}{2\log(1+(p-1)^{-1})} \cdot n^2 (1 + o_{n\to\infty}(1)) points a.a.s. intersects every such hypersurface, while o(n^2) fewer points a.a.s. misses some hypersurface. The result is applied to obtain improved lower bounds in the random Szemerédi theorem over ℝ_p^n.

Significance. If the thresholds hold, the work supplies exact leading constants for a piercing problem in algebraic combinatorics over finite fields, with the union-bound and deletion-method arguments yielding matching bounds once the super-exponentially small approximation errors are controlled. The application to random Szemerédi improves density thresholds in a concrete way.

minor comments (2)
  1. [Abstract] Clarify whether the o_{n\to\infty}(1) term in the threshold is uniform over fixed p or requires p growing slowly with n.
  2. In the statement of the general theorem for s and k, make explicit the dependence of the threshold constant on s and k.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper derives the threshold constant directly via union bound on the p^{Θ(n²)} quadratic polynomials, using the explicit per-point avoidance probability (p-1)/p + O(p^{-n/2}) for a fixed non-zero quadratic; the resulting expression log p / (2 log(1+(p-1)^{-1})) follows immediately from log(#polys) / -log(r) with no fitted parameters, self-citations, or redefinitions of inputs. The matching lower bound uses deletion or second-moment methods on the same quantities. No load-bearing step reduces to a prior result by the same authors or to a tautological fit; the argument is self-contained in the probabilistic method with p fixed and n→∞.

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

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The result is expressed in terms of the existing parameters p and n under standard asymptotic combinatorics assumptions.

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Cite this review

Pith. "Pith review of Randomly piercing algebraic sets." pith.science (2026). https://pith.science/paper/LPP7YROD

@misc{pith2026260619677,
  author       = {Pith},
  title        = {Pith review of: Randomly piercing algebraic sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LPP7YROD}},
  note         = {Machine review of arXiv:2606.19677}
}
abstract

We show, for example, that if one samples \[\frac{\log p}{2\log(1+(p-1)^{-1})} \cdot n^2(1 + o_{n\to \infty}(1))\] points in $\mathbb{F}_p^n$ at random then asymptotically almost surely this set intersects every quadratic hypersurface. We furthermore show that this is tight in that sampling $o_{n\to\infty}(n^2)$ fewer points almost surely fails to intersect some quadratic hypersurface. Our main result is a sharp threshold for the following problem: how many points in $\mathbb{F}_p^n$ does one need to randomly sample to almost surely intersect every algebraic set defined by at most $s$ polynomials each of degree at most $k$? As an application we improve lower bounds in the random Szemer\'{e}di theorem in $\mathbb{F}_p^n$, in particular obtaining a leading constant which grows as the threshold for what is considered a `dense' set in Szemer\'{e}di's theorem shrinks.

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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. Coloring sparse random Cayley graphs

    math.CO 2026-06 unverdicted novelty 7.0 of 10

    Cayley graphs on finite abelian groups with c log |Z| random generators are properly 3-colorable with high probability.

Reference graph

Works this paper leans on

20 extracted references · 2 canonical work pages · cited by 1 Pith paper

  1. [1]

    Alon,The chromatic number of random Cayley graphs, European J

    N. Alon,The chromatic number of random Cayley graphs, European J. Combin.34(2013), no. 8, 1232–

    2013

  2. [2]

    Alon and J

    N. Alon and J. H. Spencer,The probabilistic method, Fourth, Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2016. MR3524748

    2016

  3. [3]

    Altman,On Szemerédi’s theorem with differences from a random set, Acta Arith.195(2020), no

    D. Altman,On Szemerédi’s theorem with differences from a random set, Acta Arith.195(2020), no. 1, 97–108. MR4104746

    2020

  4. [4]

    Barańczuk,Reducing the number of equations defining a subset of then-space over a finite field, Ann

    S. Barańczuk,Reducing the number of equations defining a subset of then-space over a finite field, Ann. Fac. Sci. Toulouse Math. (6)33(2024), no. 1, 177–182. MR4783336

    2024

  5. [5]

    Bourgain,Ruzsa’s problem on sets of recurrence, Israel J

    J. Bourgain,Ruzsa’s problem on sets of recurrence, Israel J. Math.59(1987), no. 2, 150–166. MR920079

    1987

  6. [6]

    Briët,Subspaces of tensors with high analytic rank, Online J

    J. Briët,Subspaces of tensors with high analytic rank, Online J. Anal. Comb.16(2021), Paper No. 6,

    2021

  7. [7]

    Briët and D

    J. Briët and D. Castro-Silva,On the threshold for Szemerédi’s theorem with random differences, Electron. J. Combin.31(2024), no. 4, Paper No. 4.8, 17. MR4804109

    2024

  8. [8]

    Briët, Z

    J. Briët, Z. Dvir, and S. Gopi,Outlaw distributions and locally decodable codes, Theory Comput.15 (2019), Paper No. 12, 24. MR4028880

    2019

Show all 20 references
  1. [9]

    Briët and B

    J. Briët and B. Green,Multiple correlation sequences not approximable by nilsequences, Ergodic Theory Dynam. Systems42(2022), no. 9, 2711–2722. MR4461688

    2022

  2. [10]

    Briët and F

    J. Briët and F. Labib,High-entropy dual functions over finite fields and locally decodable codes, Forum Math. Sigma9(2021), Paper No. e19, 10. MR4228271

    2021

  3. [11]

    Christ,On random multilinear operator inequalities, 2011

    M. Christ,On random multilinear operator inequalities, 2011. arXiv:1108.5655

    2011 arXiv

  4. [12]

    Erdős and A

    P. Erdős and A. Sárközy,On differences and sums of integers. I, J. Number Theory10(1978), no. 4, 430–450. MR515054

    1978

  5. [13]

    Frantzikinakis, E

    N. Frantzikinakis, E. Lesigne, and M. Wierdl,Random differences in Szemerédi’s theorem and related results, J. Anal. Math.130(2016), 91–133. MR3574649

    2016

  6. [14]

    W. T. Gowers and J. Wolf,Linear forms and higher-degree uniformity for functions onFn p, Geom. Funct. Anal.21(2011), no. 1, 36–69. MR2773103

    2011

  7. [15]

    Lidl and H

    R. Lidl and H. Niederreiter,Finite fields, Second, Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. MR1429394

    1997

  8. [16]

    Lovett,The analytic rank of tensors and its applications, Discrete Anal

    S. Lovett,The analytic rank of tensors and its applications, Discrete Anal. (2019), Paper No. 7, 10. MR3964143

    2019

  9. [17]

    Moshkovitz and D

    G. Moshkovitz and D. G. Zhu,Quasi-linear relation between partition and analytic rank, 2024. arXiv:2211.05780

    2024

  10. [18]

    I. Z. Ruzsa,On difference sets, Studia Sci. Math. Hungar13(1978), no. 3-4, 319–326

    1978

  11. [19]

    Tung,Coloring sparse random Cayley graphs, in preparation

    N. Tung,Coloring sparse random Cayley graphs, in preparation. 19

  12. [20]

    Zheng,A note on lower bounds in Szemerédi’s theorem with random differences, Ramanujan J.69 (2026), no

    J. Zheng,A note on lower bounds in Szemerédi’s theorem with random differences, Ramanujan J.69 (2026), no. 2, Paper No. 44, 7. MR5024376 Department of Mathematics, Stanford University, CA 94305, USA Email address:daniel.h.altman@gmail.com Department of Statistics, Stanford Uni...

    2026

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