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Finite field Kakeya and Nikodym sets in three dimensions

Number theory · math.CO · math.NT · posed by Ben Lund, Shubhangi Saraf, Charles Wolf · open

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Statement

Let ϵ>0\epsilon > 0 be any constant and let qq be a sufficiently large prime power. Let LL be a set of at least q5/2+ϵq^{5/2+\epsilon} lines in Fq3\mathbb{F}_q^3 such that no plane contains more than (1/2)q3/2(1/2)q^{3/2} lines of LL. Then, ∣P(L)∣≥(1−o(1))q3|P(L)| \ge (1-o(1))q^3.

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  • Finite field Kakeya and Nikodym sets in three dimensions
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh, with Shengtong Zhang

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed statement: for every fixed ϵ>0\epsilon>0, all sufficiently large prime powers qq, and every set LL of affine lines in Fq3\mathbb F_q^3 with

    ∣L∣≥q5/2+ϵand∣{ℓ∈L:ℓ⊂π}∣≤12q3/2|L|\ge q^{5/2+\epsilon} \quad\text{and}\quad |\{\ell\in L:\ell\subset \pi\}|\le \tfrac12 q^{3/2}

    for every affine plane π\pi, the union P(L)=⋃ℓ∈LℓP(L)=\bigcup_{\ell\in L}\ell satisfies

    ∣P(L)∣≥(1−o(1))q3.|P(L)|\ge (1-o(1))q^3 .

    This is the literal Conjecture 5 stated in the input.

    Result: The conjecture is false. In fact, for every fixed 0<ϵ<1/20<\epsilon<1/2 and infinitely many qq, there are such line sets whose union has asymptotic density 1/21/2.

    Let qq be odd and let δ∈Fq×\delta\in\mathbb F_q^\times be a nonsquare. Consider the elliptic paraboloid

    S={(a,b,a2−δb2):a,b∈Fq}.S=\{(a,b,a^2-\delta b^2):a,b\in\mathbb F_q\}.

    For p=(a,b,a2−δb2)∈Sp=(a,b,a^2-\delta b^2)\in S and [u:v]∈P1(Fq)[u:v]\in\mathbb P^1(\mathbb F_q), the tangent line at pp in projected direction [u:v][u:v] is

    ℓa,b,u,v={(a+tu, b+tv, a2−δb2+2t(au−δbv)):t∈Fq}.\ell_{a,b,u,v} =\{(a+tu,\ b+tv,\ a^2-\delta b^2+2t(au-\delta bv)):t\in\mathbb F_q\}.

    Since δ\delta is nonsquare, u2−δv2≠0u^2-\delta v^2\neq0 for every [u:v][u:v]. Let LL be the set of those tangent lines with u2−δv2u^2-\delta v^2 a square.

    Among the q+1q+1 directions [u:v][u:v], exactly (q+1)/2(q+1)/2 have u2−δv2u^2-\delta v^2 square, so

    ∣L∣=q2q+12.|L|=q^2\frac{q+1}{2}.

    Thus ∣L∣≥q5/2+ϵ|L|\ge q^{5/2+\epsilon} for every fixed ϵ<1/2\epsilon<1/2 and all sufficiently large odd qq.

    No affine plane contains too many of these lines. Indeed, a plane intersects SS in at most q+1q+1 points, except that if it is tangent at some pp, it contains only that point of SS but may contain all selected tangent directions at pp, namely (q+1)/2(q+1)/2 lines. Hence every plane contains at most q+1q+1 lines of LL, which is <12q3/2<\tfrac12 q^{3/2} for all sufficiently large qq.

    Now define

    Q(x,y,z)=x2−δy2−z.Q(x,y,z)=x^2-\delta y^2-z.

    For a point on ℓa,b,u,v\ell_{a,b,u,v} with parameter tt,

    Q=t2(u2−δv2),Q=t^2(u^2-\delta v^2),

    so QQ is either 00 or a square. Conversely, if Q(x,y,z)=r2Q(x,y,z)=r^2, then the direction [1:0][1:0] line tangent at

    (x−r, y, (x−r)2−δy2)(x-r,\ y,\ (x-r)^2-\delta y^2)

    contains (x,y,z)(x,y,z). Therefore

    P(L)={(x,y,z):Q(x,y,z)∈{0}∪(Fq×)2}.P(L)=\{(x,y,z): Q(x,y,z)\in \{0\}\cup(\mathbb F_q^\times)^2\}.

    For each pair (x,y)(x,y), exactly (q+1)/2(q+1)/2 choices of zz satisfy this. Hence

    ∣P(L)∣=q2q+12=(12+o(1))q3,|P(L)|=q^2\frac{q+1}{2} =\left(\frac12+o(1)\right)q^3,

    not (1−o(1))q3(1-o(1))q^3. This disproves the conjecture.

    Citation: No prior resolution is used here. The conjecture is from Lund–Saraf–Wolf, “Finite field Kakeya and Nikodym sets in three dimensions,” SIAM J. Discrete Math. 32 (2018), arXiv:1609.01048.

    • Curated solution (PDF) — not hosted yetpdf

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification KNOWN

    PASS

    The solution attacks the exact conjecture and gives a valid counterexample for any fixed 0<ϵ<1/20<\epsilon<1/2 along all large odd qq. The tangent-line construction has

    ∣L∣=q2(q+1)/2≥q5/2+ϵ,|L|=q^2(q+1)/2\ge q^{5/2+\epsilon},

    while every affine plane contains at most q+1=o(q3/2)q+1=o(q^{3/2}) of the selected lines, so the plane condition is satisfied.

    The union computation is correct: along each selected tangent line,

    Q=x2−δy2−zQ=x^2-\delta y^2-z

    takes only values 00 or nonzero squares, and conversely every point with QQ square/zero lies on a selected tangent line. Hence

    ∣P(L)∣=q2q+12=(1/2+o(1))q3,|P(L)|=q^2\frac{q+1}{2}=(1/2+o(1))q^3,

    contradicting the conjectured (1−o(1))q3(1-o(1))q^3. I found no fatal gap or mismatch with the stated conjecture.

    Novelty assessment

    KNOWN

    Classification rationale: The counterexample is an affine translation of a standard finite-geometry construction: tangent lines to an elliptic quadric/paraboloid over Fq\mathbb F_q, split into the two quadratic-character tangent classes. One tangent class covers precisely the square-value side of the associated quadratic form, giving the same asymptotic half-density union and the same small plane intersections. Thus the construction is not genuinely new, even if it may not have been explicitly advertised as a disproof of Lund–Saraf–Wolf Conjecture 5.

    Literature check: I checked the original Lund–Saraf–Wolf paper and its citation graph, including later finite-field Kakeya/Nikodym papers, and found no explicit statement “Conjecture 5 is false.” However, searches around elliptic quadrics, tangent lines, Cameron–Liebler line classes, and finite-field paraboloid tangent constructions led to the classical Bruen–Drudge construction. Their elliptic-quadric tangent-line partition is essentially the projective version of the prover’s construction; deleting the point/lines at infinity gives exactly the affine counterexample.

    Citation: A. A. Bruen and K. Drudge, “The construction of Cameron–Liebler line classes in PG(3,q)\mathrm{PG}(3,q),” Finite Fields and Their Applications 5(1):35–45, 1999. Also: B. Lund, S. Saraf, C. Wolf, “Finite field Kakeya and Nikodym sets in three dimensions,” SIAM J. Discrete Math. 32 (2018), 2836–2849.

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