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arxiv: 2606.19677 · v1 · pith:LPP7YRODnew · submitted 2026-06-18 · 🧮 math.NT · math.CO

Randomly piercing algebraic sets

Daniel Altman , Nathan Tung This is my paper

Pith reviewed 2026-06-26 16:08 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords algebraic setsfinite fieldsquadratic hypersurfacesprobabilistic methodthreshold phenomenaSzemerédi theoremrandom samplingpiercing
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The pith

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.

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

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 rac{\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.

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 rac{\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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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 rac{\log p}{2 \log(1+(p-1)^{-1})}.

read the original 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.

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.

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

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→∞.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 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.

pith-pipeline@v0.9.1-grok · 5704 in / 1158 out tokens · 34513 ms · 2026-06-26T16:08:19.455378+00:00 · methodology

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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.

  1. Coloring sparse random Cayley graphs

    math.CO 2026-06 unverdicted novelty 7.0

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

Reference graph

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