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Bisector energy and pinned distances in positive characteristic

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Pith's one-line read This paper proves that in a finite-field plane, any sufficiently small point set that is not mostly contained in a zero-distance line has a point determining Ω(|A|^{2/3}) distinct distances to the rest of the set.

desk verdict New pinned-distance exponent 2/3 for small sets in F_q^2; the proof is sound and the one flagged gap is fillable by a short KST argument. read the letter →

arxiv 1908.04618 v3 pith:DJCUVC7T submitted 2019-08-13 math.CO

Brendan Murphy , Misha Rudnev , Sophie Stevens This is my paper · ORCID
classification math.CO MSC 52C10
keywords pinneddistancesfinitefieldsbisectorenergyisoscelestrianglespoint-planeincidenceskinematicmappingpositivecharacteristicdistinct
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 establishes a new lower bound on pinned distances in finite fields: if a point set A in $F_q^{2}$ has size at most $p^{{4/3}}$, where p is the characteristic, and no isotropic line—a line on which the quadratic distance always vanishes—carries more than a third of the points, then some point of A determines ≫ |A|^{2/3} distinct distances to the other points. This improves the previous 20/37 exponent for this regime. The route is indirect: the proof bounds the number of non-degenerate isosceles triangles in A, because many pinned distances force few repeated distance pairs and hence few isosceles triangles. To get the triangle bound, rigid motions of the plane are embedded into projective 3-space, turning the count of axially symmetric same-length segments into a point-plane incidence count. The result matters because distinct distances in finite fields is the finite analogue of a central open problem in discrete geometry, and it tightens the threshold question of when a set is large enough to determine every possible distance.

What carries the argument

The mechanism is the modified bisector energy B^*(A), which counts pairs of ordered point-pairs in A whose perpendicular bisectors coincide, but only through points off the bisector line; this quantity controls isosceles triangles by T^*(A) ≤ |A| B^*(A)^{1/2}. To bound B^*(A), the paper uses the Blaschke-Grünwald kinematic mapping—an embedding of the group of rigid motions of $F^{2}$ into an open subset of projective 3-space, proved here for arbitrary fields via Clifford algebras. Under this embedding, the set of segments of a fixed nonzero length r becomes a set of points, and the axial symmetries that pair them become planes; the number of paired segments is exactly the number of point-plane incidences. A point-plane incidence bound for $F^{3}$, applicable when the number of planes is ≪ $p^{2}$, then gives B^*(A) ≪ M|A|^2 + |A| Q^*(A)^{1/2}, where M is the maximum number of collinear or co-circular points; a pruning lemma for rich lines and circles removes the M-dependence and yields the isosceles-triangle bound.

What would settle it

For an odd prime p, take a set A ⊂ $F_p^{2}$ with |A| ≈ $p^{{4/3}}$ and at most a third of A on any isotropic line, and compute max_{a∈A} |{d(a,b) : b∈A}| and the isosceles-triangle count T^*(A). A family with max_a |Δ(A,a)| = o(|A|^{2/3}), or with T^*(A) growing faster than K|A|^{7/3} for every fixed K, would refute the paper's central claims. A cheaper target: find a fixed nonzero distance r whose repetition count |S_r| exceeds C|A|^{3/2}; that breaks the $p^{2}$-condition on which the incidence step rests.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a bound on the number of isosceles triangles rather than a direct distance count. For any A ⊂ $F^{2}$ with |A| ≤ $p^{{4/3}}$ in characteristic p and with at most a third of its points on a single isotropic subspace, the number T^*(A) of non-degenerate isosceles triangles with nonzero equal side lengths is O(|A|^{7/3}). A second-moment inequality then converts this into the pinned-distance statement: some a ∈ A determines Ω(|A|^{2/3}) distinct nonzero distances to other points of A. The same triangle bound yields Q^*(A) ≪ |A|^{10/3}, a bound on nonzero distance quadruples, and together with known large-set results it improves the previously known lower bounds on distinct distances over finite fields.

Load-bearing premise

The proof's gate is the unproved distance-repetition bound |S_r| ≪ |A|^{3/2} for each fixed nonzero distance r, assumed to hold over arbitrary fields; if some finite-field set has far more pairs at one distance than this, the $p^{{4/3}}$ hypothesis no longer guarantees the incidence bound applies and the main theorem collapses.

Editorial extensions

If this is right

  • A set A satisfying the hypotheses has a point whose distance set to A has at least c|A|^{2/3} elements; hence the total number of distinct distances determined by A is also ≫ |A|^{2/3}.
  • The number of non-degenerate isosceles triangles in A is O(|A|^{7/3}), a structural bound that holds even when A contains rich lines or rich circles.
  • The number of nonzero distance quadruples Q^*(A) is O(|A|^{10/3}).
  • Together with the Fourier and spectral results for large sets in F_q^2, the small-set bound gives the best current lower bounds for distinct distances over finite fields across all sizes.
  • The theorem applies to arbitrary fields, not just prime fields: for any field F of positive characteristic p, any set A with |A| ≤ p^{4/3} and the isotropic-line condition has the stated pinned-distance lower bound.

Reading between the lines

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

  • The isosceles-triangle estimate is likely the reusable output: any two-point statistic in F^2 that can be carried through the kinematic embedding should inherit an essentially cubic-root improvement from the point-plane incidence bound, so similar exponents may hold for Minkowski distances or k-simplex energies in the same size range.
  • The imported distance-repetition estimate is the natural place to look for a counterexample; a set with unusually many pairs at one nonzero distance would invalidate the p^{4/3} threshold even if the final theorem happens to be true.
  • For Cartesian products A = X × X one would expect the same machinery to do better than |A|^{2/3}, since product structure should suppress coincident bisectors; the energy quantity Q^*(A) isolated in the proof is the object that would control such an improvement.
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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

3 major / 7 minor

Summary. The paper proves a new lower bound for pinned distances over finite fields and, more generally, arbitrary fields. Its main result, Theorem 2, states that if A is a subset of F^2 with at most a third of its points on any isotropic line and |A| is at most p^{4/3} in characteristic p, then there is a point in A determining ≫ |A|^{2/3} distinct distances. The proof introduces a modified bisector energy B*(A), relates it to the number T*(A) of non-degenerate isosceles triangles (Lemma 7), and bounds T*(A) by |A|^{7/3} (Proposition 9) using a point-plane incidence theorem after embedding segments into projective three-space via the Blaschke-Grünwald kinematic mapping. Proposition 8 bounds bisector energy in terms of distance quadruples, and Proposition 9 removes the dependence on rich lines and circles by a pruning argument. An appendix gives a Clifford-algebra derivation of the kinematic mapping over arbitrary fields.

Significance. If correct, Theorem 2 improves the previous best pinned-distance exponent 20/37 due to Lund and Petridis to 2/3 for sets of size up to p^{4/3}, and it yields a new upper bound on the number of isosceles triangles. The reduction of bisector energy to point-plane incidences via the kinematic mapping is elegant and likely to be useful; the Clifford-algebra appendix extends classical kinematics to arbitrary fields. The derivation has no free parameters and rests on a published incidence theorem, and the bootstrap through equation (9) is a genuine idea. These are real strengths. The main caveats are several local gaps and small errors in the written proof, all of which appear repairable without changing the central argument.

major comments (3)
  1. [Section 3, Lemma 6] The lemma is stated under the hypothesis that at most M points of A lie on a line, but Theorem 2 only assumes a bound on points on isotropic lines. As written, the application of Lemma 6 to Theorem 2 is not justified when A has many points on a non-isotropic line, because the proof's lower bound |A \ (a+C_0)| >= |A| - 2M + 1 becomes vacuous if M is taken as the maximum over all lines. The proof only needs the bound for the two isotropic lines comprising a+C_0, so the lemma should be restated with M denoting the maximum number of points on an isotropic line, or the proof of Theorem 2 should be modified to handle non-isotropic collinear structure separately.
  2. [Section 3, Eq. (9)] The displayed chain ending in = |A|T*(A) is incorrect: by equation (6), the double sum equals T*(A) + |A|^2, not T*(A). The correct estimate is Q*(A) <= |A|(T*(A) + |A|^2). This still suffices for the bootstrap in Proposition 9 because the extra |A|^3 term is of lower order than |A|^{7/3}, but the displayed equality must be corrected.
  3. [Section 4, Claim 1] The estimate |S_r| << |A|^{3/2} is cited to Erdős [8] and is used to verify the hypothesis |Π| << p^2 of the point-plane incidence theorem (Theorem 5). Since [8] is a statement about the real plane and the present argument is over arbitrary fields, the authors should either prove the bound with a short Kővári-Sós-Turán argument, using that two distinct nonzero-radius circles meet in at most two points, or provide a reference that covers arbitrary fields. Without this justification, the passage from |A| <= p^{4/3} to the incidence-theorem hypothesis is unsupported.
minor comments (7)
  1. [Abstract] The abstract credits the point-plane incidence theorem to 'the third author'; the theorem is attributed in the body to the second author (Rudnev), and this should be corrected.
  2. [Abstract] The phrase 'improves all previously known lower bounds on distinct distances over finite fields' is stronger than what is shown: the improvement applies in the range |A| <= p^{4/3}, and for F_q with q > p there is a gap before the known large-set results apply.
  3. [Section 3, Lemma 6 proof] The display '|A|(|A|-2M+1) <= |A||A \ (a+C_0)| = sum_{a in A} ...' has a free variable a in the middle expression and appears to be missing a summation; it should read sum_{a in A} |A \ (a+C_0)|.
  4. [Section 4, Claim 1] The notation τ(A) is used without definition; the line ℓ_τ is mentioned, but the reflection τ is not introduced.
  5. [Section 4, Claim 1] The assertion that one can choose ℓ_τ so that g^{-1}h has no fixed points on ℓ_τ for all g,h in G_r is impossible when g=h, since the identity fixes every line; the distinctness argument for the planes Π should exclude the diagonal case.
  6. [Section 5, Lemma 12] Lemma 12 is stated with T(A) but the proof concerns T*(A), and 'coplanar' should read 'co-circular' in the statement of the pruning lemma.
  7. [Section 5] The inequality 'sum_ℓ i_{A'}(ℓ)b*_{A'}(ℓ) <= 2|A'|^2 + |A'|B*(A')^{1/2}' is not a direct consequence of Lemma 7 as stated; please clarify the derivation, since Lemma 7 gives T*(A') <= |A'|B*(A')^{1/2}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pinned-distance bound follows from a genuine bootstrap on bisector energy, with only independent external incidence and distance-repetition inputs.

full rationale

The paper's central derivation chain is not circular. Proposition 9 bounds the number of non-degenerate isosceles triangles T*(A) by |A|^{7/3} via Lemma 7 (T* ≤ |A| B*^{1/2}), Proposition 8 (B* ≪ M|A|^2 + |A| Q*^{1/2}), and equation (9) (Q* ≤ |A|T*). After pruning heavy lines/circles, one obtains T*(A) ≪ |A|^{7/3} + |A|^{7/4} T*(A)^{1/4}, which is a legitimate self-improving inequality and is not an identity: substituting x = T*/|A|^{7/3} gives x ≤ C + Cx^{1/4}, so x is bounded. The point-plane incidence theorem (Theorem 5) is due to one of the authors, but it is a published, parameter-free external theorem with hypotheses (|P| ≪ p^2 and collinearity parameter k) that do not include the pinned-distance conclusion; citing it is not circular. The auxiliary bound |S_r| ≪ |A|^{3/2} cited to Erdős is not proved in the paper, but it is a standard codegree/Kővári–Sós–Turán fact over any field and is not a restatement of the target result. The Blaschke–Grünwald embedding is proved in Appendix A rather than assumed. No parameter in the proof is fitted to the quantity being predicted, and no quantity being 'predicted' is an input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no speculative physical entities. It relies on three external results: Rudnev's point-plane incidence theorem, the Erdős distance-repetition bound, and Lund and Petridis's structural lemma about axially symmetric segments. The algebraic closure step and the choice of the axis ℓ_τ are justified in the text but rely on standard facts about projective spaces and rigid motions.

assumptions (5)
  • domain assumption Rudnev point-plane incidence theorem: for point set P and plane set Π in F^3, with |P| ≤ |Π| and |P| ≪ p^2 in positive characteristic, I(P,Π) ≪ |P|^{1/2}|Π| + k|Π|, where k is the maximum collinear points in P.
    Invoked in Claim 1 to bound the incidence count I(S_r, A(S_r)); accepted external theorem, not proved in this paper.
  • domain assumption For a fixed nonzero distance r, the number of segments S_r in A × A satisfies |S_r| ≪ |A|^{3/2}.
    Used in Claim 1 to guarantee |Π| = |S_r| ≤ p^2 for the incidence theorem; cited to Erdős [8] without proof for finite fields.
  • domain assumption Lund-Petridis structural lemma: endpoints of every segment axially symmetric to two fixed segments of length r lie on a pair of concentric circles or parallel lines.
    Used to bound the maximum number of collinear points in P and planes in Π by M, so that k ≤ M in Theorem 5.
  • domain assumption Passing from F to its algebraic closure preserves the quantities T^*, B^*, Q^*, and the incidence counts for A.
    Claim 1 asserts WLOG F is algebraically closed; plausible for F-rational configurations, stated without detailed proof.
  • standard math Over an algebraically closed field one can choose a non-isotropic line ℓ_τ avoiding the fixed points of the finitely many non-identity products g^{-1}h.
    Justifies the choice of τ so that distinct segments in G_r give distinct planes; standard projective geometry.

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Pith. "Pith review of Bisector energy and pinned distances in positive characteristic." pith.science (2026). https://pith.science/paper/DJCUVC7T

@misc{pith2026190804618,
  author       = {Pith},
  title        = {Pith review of: Bisector energy and pinned distances in positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJCUVC7T}},
  note         = {Machine review of arXiv:1908.04618}
}
abstract

We prove a new lower bound for the number of pinned distances over finite fields: if $A$ is a sufficiently small subset of $\mathbb{F}_q^2$, then there is an element in $A$ that determines $\gg |A|^{2/3}$ distinct distances to other elements of $A$. Combined with results for large subsets $A\subseteq\mathbb{F}_q^2$, this improves all previously known lower bounds on distinct distances over finite fields. In fact, we obtain an upper bound for the number of isosceles triangles determined by $A$. For that we use the concept of bisector energy. It turns out that the latter can be expressed as a point-plane incidence bound, so one can use a theorem of the third author. The conversion to this incidence problem relies on the Blaschke-Gr\"unwald kinematic mapping -- an embedding of the group of rigid motions of $\mathbb{F}_q^2$ into an open subset of the projective three space. This has long been known in kinematics and geometric algebra; we provide a proof for arbitrary fields using Clifford algebras.

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