Spherical restriction R^*_{S_j}(2 o r)≲1 holds for all r>23/7 over F_p^4, yielding the first improvement past the (d+1)/2 barrier for the four-dimensional prime-field Erdős–Falconer distance problem.
A short proof of R udnev’s point-plane incidence bound
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abstract
In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a three-dimensional space, where it applies a bound of Guth and Katz on intersection points of lines. We describe a simple geometric map that directly sends point-plane incidences to line-line intersections in space, allowing us to reprove Rudnev's theorem with fewer technicalities.
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2026 2representative citing papers
Entropy lower bounds are established for sums and products, including a max(H(X+X'), H(XX')) bounded below by a linear function of H(X) and min-entropy of X over arbitrary fields.
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On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications
Spherical restriction R^*_{S_j}(2 o r)≲1 holds for all r>23/7 over F_p^4, yielding the first improvement past the (d+1)/2 barrier for the four-dimensional prime-field Erdős–Falconer distance problem.
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Entropy lower bounds and sum-product phenomena
Entropy lower bounds are established for sums and products, including a max(H(X+X'), H(XX')) bounded below by a linear function of H(X) and min-entropy of X over arbitrary fields.