Finite field Kakeya and Nikodym sets in three dimensions
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Statement
Let be any constant and let be a sufficiently large prime power. Let be a set of at least lines in such that no plane contains more than lines of . Then, .
Context
Candidate 3 of the open problems stated in "Finite field Kakeya and Nikodym sets in three dimensions", extracted for the Scalable Mathematical Discovery run.
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NEW
Problem: Reconstructed statement: for every fixed , all sufficiently large prime powers , and every set of affine lines in with
for every affine plane , the union satisfies
This is the literal Conjecture 5 stated in the input.
Result: The conjecture is false. In fact, for every fixed and infinitely many , there are such line sets whose union has asymptotic density .
Let be odd and let be a nonsquare. Consider the elliptic paraboloid
For and , the tangent line at in projected direction is
Since is nonsquare, for every . Let be the set of those tangent lines with a square.
Among the directions , exactly have square, so
Thus for every fixed and all sufficiently large odd .
No affine plane contains too many of these lines. Indeed, a plane intersects in at most points, except that if it is tangent at some , it contains only that point of but may contain all selected tangent directions at , namely lines. Hence every plane contains at most lines of , which is for all sufficiently large .
Now define
For a point on with parameter ,
so is either or a square. Conversely, if , then the direction line tangent at
contains . Therefore
For each pair , exactly choices of satisfy this. Hence
not . 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.
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0 human reviews · 1 machine checkNo person has reviewed this attempt. 1 machine check below — a machine check is not human verification.
Machine check · not human verification
machine: correctscope Full solution as submitted; SMD novelty classification KNOWN
PASS
The solution attacks the exact conjecture and gives a valid counterexample for any fixed along all large odd . The tangent-line construction has
while every affine plane contains at most of the selected lines, so the plane condition is satisfied.
The union computation is correct: along each selected tangent line,
takes only values or nonzero squares, and conversely every point with square/zero lies on a selected tangent line. Hence
contradicting the conjectured . 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 , 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 ,” 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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