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The sharp exponent for the minimal distance problem

T0 review · 1 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The minimal distance problem has sharp exponent 2/3: for every ε, n points and lines can be kept at least n^{-2/3-ε} apart, matching the best possible upper bound.

desk verdict The paper solves the minimal distance problem with a genuinely new trace-zero lattice construction; the proof is sound and the finite-field corollary is a real bonus, though the sharp exponent still leans on the cited upper bound. read the letter →

arxiv 2607.20422 v2 pith:ASEFE3C7 submitted 2026-07-22 math.CO math.MGmath.NT

Cosmin Pohoata This is my paper · ORCID
classification math.COmath.MGmath.NT MSC 52C1011R04
keywords minimaldistanceproblempoint-lineincidencesHeilbronntriangletrace-zerolatticetotallyrealnumberfieldsinducedmatchingsfinitesquare-difference-freesets
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 the exact asymptotic scale for the minimal distance problem: among n point-line pairs in the unit square, the largest possible minimum distance between a point and a non-assigned line is n^{-2/3+o(1)}. The new contribution is a lower-bound construction achieving n^{-2/3-ε} for every ε>0, using a trace-zero lattice inside a totally real number field instead of the previous integer-based square-difference-free sets. Combined with a previously known upper bound, this closes the gap and settles the exponent. The same construction yields induced point-line matchings of size roughly q^{3/2-ε} in F_q^2 for a positive density of primes q, disproving a conjecture that only q^{3/2-c} was possible.

What carries the argument

The key object is the trace-zero lattice Λ0_K = {a ∈ 2O_K : Tr_{K/Q}(a)=0} in a totally real number field K of degree d. Positive definiteness of the quadratic form z ↦ Tr(z^2) guarantees that this set is square-difference-free: if a-a'=z^2 with a,a' in the lattice, then z=0. This yields far more elements than integer square-difference-free sets, and the field norm of the incidence-detecting quantity D(p,p') gives quantitative separation after one real embedding. The field is chosen to be the maximal real subfield of a cyclotomic field, of degree d=(r-1)/2, so the resulting exponent is 2/3 + 4/(9d-6), which tends to 2/3.

What would settle it

Find an infinite sequence of n together with point-line configurations in [0,1]^2 for which min_{i≠j} dist(x_i, ℓ_j) ≥ n^{-2/3+ε} for some fixed ε>0; this would contradict the claimed upper bound. Alternatively, exhibit a nonzero element z of the trace-zero lattice with Tr(z^2)=0, which would break the square-difference-free property.

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Extended reading notes

Core claim

The central claim is that the minimal distance exponent is exactly 2/3. The proof constructs, for every n, a configuration with off-diagonal distances at least n^{-2/3-ε}. The mechanism is to replace the parabola-level-set construction over the integers, whose separation was limited by square-difference-free sets of size roughly N^{0.733...}, by the same construction over a fixed totally real number field K of high degree. The replacement set is the trace-zero lattice Λ0_K ⊂ 2O_K, whose elements a satisfy Tr(a)=0. Because the trace form Tr(z^2) is positive definite, no two distinct trace-zero elements differ by a square; this gives the algebraic square-difference-free property with many more

Load-bearing premise

The sharp exponent 2/3 relies on a previously established upper bound Δ_PL(n) ≤ n^{-2/3+o(1)} from an earlier paper; the new lower-bound construction stands on its own, but the resolution of the problem requires that upper bound to be correct.

Editorial extensions

If this is right

  • The minimal distance problem is resolved: Δ_PL(n) = n^{-2/3+o(1)}.
  • The current best upper bound for the triangle-area problem, n^{-7/6+o(1)}, follows from this exponent; the paper conjectures further improvement to n^{-7/6-c}.
  • For a positive density of primes q, F_q^2 contains induced point-line matchings of size q^{3/2-ε}; hence the supremum limit of log IM(2,q)/log q over primes is 3/2, ruling out any power saving below 3/2 for all large primes.
  • The construction yields a number-field generalization of the square-difference problem with bounds X^{d-1} ≲ s_K(X) ≲ X^d exp(-c√log X).

Reading between the lines

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

  • The trace-zero method may transfer to higher-dimensional incidence problems, where a codimension-one restriction dilutes a fixed obstruction in a similar way.
  • The same construction could yield improved lower bounds for the square-difference problem in number fields, or for finite-field analogues, by choosing fields with many split primes and building small Nikodym sets from the induced matchings.
  • The role of the high-degree field here is opposite to most recent constructions: the degree is used to dilute a fixed codimension rather than to amplify a local gain, which might be a reusable template.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper resolves the exponent in the minimal distance problem by constructing, for every fixed ε>0 and all sufficiently large n, n point–line pairs (x_i,ℓ_i) in [0,1]^2 with x_i∈ℓ_i and dist(x_i,ℓ_j)≥n^{-2/3−ε} for i≠j. The construction works in a totally real number field K of large degree: the trace-zero lattice Λ^0_K={a∈2O_K: Tr_{K/Q}(a)=0} is square-difference-free because Tr(z^2)=Σσ(z)^2>0, and the norm bound (30) converts nonvanishing of D(p,p') into Euclidean separation. Balancing R=M^2 gives size ≍_K M^{3d−2} and separation ≍_K M^{−2d}, yielding exponent 2d/(3d−2)=2/3+4/(9d−6), which tends to 2/3. Together with the known upper bound Δ_PL(n)≤n^{−2/3+o(1)} from [7], this gives Δ_PL(n)=n^{−2/3+o(1)}. The same integral construction, reduced modulo split primes q≡±1 mod r, gives induced point–line matchings in F_q^2 of size ≳_r q^{3/2−2/(r−1)}, disproving a conjecture of Hunter–Pohoata–Verstraëte–Zhang.

Significance. If correct, this is a definitive resolution of a central exponent in combinatorial geometry. The new lower bound is self-contained and introduces a clean number-field mechanism that bypasses the Furstenberg–Sárközy/Ruzsa barrier; the proofs of Proposition 3.2, Proposition 4.1 and Proposition 5.1 are explicit and checkable, and the algebra in the distance formula (31) is correct. The finite-field corollary is surprising and strong, giving a positive-density set of primes where IM(2,q) is within q^{o(1)} of the trivial q^{3/2} bound. The only external input to the sharp exponen statement is the upper bound (2) from [7]; this is a normal reliance on a published theorem, but Corollary 1.2 should explicitly flag that dependency.

major comments (1)
  1. [§4, proof of Theorem 1.1] The proof applies Corollary 4.2 and obtains configurations of size N_M ≍_K M^{3d−2}, not of every prescribed size n. As written, this proves Δ_PL(N_M) ≥ N_M^{−2/3−ε} only along the sequence N_M; it does not establish the theorem's 'for every integer n≥n0(ε)' statement, which Corollary 1.2 needs. The gap is easily repaired: for each large n choose M with N_M between n and Cn (possible since consecutive values of M^{3d−2} are at ratio 1+o(1)), then pass to a subset of n of the point–line pairs; the separation lower bound is preserved. Please add this argument.
minor comments (4)
  1. [Equation (25)] The definition of N is ambiguous: it should read N=(R+M^2)/2, not R+M^2/2. The proof uses the parenthesized version.
  2. [§5] The rational prime q and the prime ideal q are both denoted q; use a different symbol (e.g. fraktur q) to avoid confusion.
  3. [Corollary 1.2] State explicitly that the upper bound is the cited inequality (2) from [7] and is not re-proved here; this makes the dependency of the sharp-exponent claim transparent.
  4. [§4, Corollary 4.2] The displayed lower bound contains a factor 3/(2√2); a one-line derivation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the new lower-bound construction is self-contained, and the cited upper bound used for the sharp exponent is external prior work.

full rationale

The paper's central new result is Theorem 1.1, a lower bound Δ_PL(n) ≥ n^{-2/3−ε}. Its proof is self-contained: Section 3 defines the trace-zero lattice Λ0_K, proves square-difference-freeness from the positive definiteness of Tr(z^2) (Prop. 3.2), and Section 4 derives the Euclidean separation via the norm bound (30) and the distance identity (31). The parameter balance R = M^2 is a free choice, and the exponent 2d/(3d−2) follows explicitly from the counts |A_K(R)| ≍ R^{d−1}, |Y_K(M)| ≍ M^d. No fitted quantity is renamed as a prediction. Corollary 1.2 combines this lower bound with the upper bound (2), cited from Cohen–Pohoata–Zakharov [7]; that is a previously published, parameter-free result, not a consequence of the present construction, so relying on it is normal external support rather than circularity. The finite-field Theorem 1.3 is likewise derived from the same explicit construction: Proposition 5.1 reduces it modulo a completely split prime and uses only standard norm/ideal facts, and the set of primes is produced by the prime number theorem in arithmetic progressions. No step assumes the conclusion, defines a quantity in terms of the target, or imports a uniqueness theorem from the authors' prior work. Section 2 re-derives the relevant machinery from [12] rather than importing it as a black box. The only minor omission is an explicit subset argument to pass from sizes ≍ M^{3d−2} to every n, but this is a routine technical step and not a circular reduction.

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

The central construction uses standard algebraic number theory and lattice facts; the prior upper bound is an external published result. There are no fitted data parameters and no new physical or postulated entities. The auxiliary quantities d, R, M are proof parameters chosen by the argument, not fitted values.

assumptions (6)
  • standard math For K=Q(ζ_p+ζ_p^{-1}), K is totally real of degree (p−1)/2 and the trace form q_K(z)=Σσ_j(z)^2 is positive definite.
    Invoked in Section 4 to prove Theorem 1.1; standard cyclotomic field fact.
  • standard math For a lattice Λ⊂R^m of rank r, |Λ∩[−T,T]^m| ≍_Λ T^r.
    Used in Proposition 3.2 to estimate |A_K(R)| and |Y_K(M)|.
  • standard math For a nonzero algebraic integer α, |N_{K/Q}(α)| ≥ 1.
    Used in (30) and (33) to turn nonvanishing into Euclidean or modular separation.
  • standard math In a Dedekind domain, if a prime ideal q contains α, then q divides the principal ideal (α) and N(q) divides |N(α)|.
    Used in Proposition 5.1 to rule out off-diagonal incidences after reduction modulo q.
  • standard math For K_r=Q(ζ_r+ζ_r^{-1}), a rational prime q≠r splits completely iff q≡±1 mod r; Dirichlet's theorem gives positive relative density.
    Used in Theorem 1.3 to choose the positive-density set of primes q.
  • domain assumption The prior upper bound Δ_PL(n) ≤ n^{-2/3+o(1)} of Cohen–Pohoata–Zakharov [7] is correct.
    Required for Corollary 1.2; not re-derived in this paper.

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Cite this review

Pith. "Pith review of The sharp exponent for the minimal distance problem." pith.science (2026). https://pith.science/paper/ASEFE3C7

@misc{pith2026260720422,
  author       = {Pith},
  title        = {Pith review of: The sharp exponent for the minimal distance problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ASEFE3C7}},
  note         = {Machine review of arXiv:2607.20422}
}
abstract

We show that for every fixed $\varepsilon>0$, there exist arbitrarily large families of point-line pairs $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$, with $x_i \in \ell_i$ for all $i$, and such that $\operatorname{dist}(x_i,\ell_j)\ge n^{-2/3-\varepsilon}$ for all $i \neq j$. Combined with a previous result of Cohen, the author, and Zakharov, this solves the minimal distance problem. The same construction also comes with an unexpected finite field consequence: for every $\varepsilon>0$, there exists a set of primes $q$ of positive relative density for which $\mathbb F_q^2$ contains an induced point-line matching of size $\gtrsim q^{3/2-\varepsilon}$. This disproves a conjecture of Hunter, the author, Verstra\"ete and Zhang.

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