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We establish new linear and trilinear bounds for collections of tubes inR   4    that satisfy the polynomial Wolff axioms. In brief, a collection of   δ  ‐tubes satisfies the Wolff axioms if not too many tubes can be contained in the   δ  ‐neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the   δ  ‐neighborhood of a low degree algebraic variety.  First, we prove that if a set of    δ   −  3tubes inR   4    satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least    δ   1  −  1  /  28    . We also prove a more technical statement which is analogous to a maximal function estimate at dimension    3  +  1  /  28   . Second, we prove that if a collection of    δ   −  3tubes inR   4    satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least    δ   3  /  4    . Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension    3  +  1  /  4   .  We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension    3  +  1  /  28   , and in particular this implies that every Kakeya set inR   4    must have Hausdorff dimension at least    3  +  1  /  28   . This would be an improvement over the current best bound of 3, which was established by Wolff in 1995.

Polynomial Wolff axioms and Kakeya‐type estimates in R 4