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On the polynomial Wolff axioms

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We confirm a conjecture of Guth concerning the maximal number of \({\delta}\)-tubes, with \({\delta}\)-separated directions, contained in the \({\delta}\)-neighborhood of a real algebraic variety. Modulo a factor of \({\delta^{-{\varepsilon}}}\), we also prove Guth and Zahl’s generalized version for semialgebraic sets. Although the applications are to be found in harmonic analysis, the proof will employ deep results from algebraic and differential geometry, including Tarski’s projection theorem and Gromov’s algebraic lemma.

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The first author would like to thank Josh Zahl for helpful discussions. In particular the proof of Lemma 2.2 came from a conversation with him. The second author would like to thank Jonathan Hickman for helpful discussions regarding the application to restriction.

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Correspondence to Keith M. Rogers.

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Supported by NSF grant DMS 1565904 and by MINECO Grants SEV-2015-0554 and MTM2017- 85934-C3-1-P.

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Katz, N.H., Rogers, K.M. On the polynomial Wolff axioms. Geom. Funct. Anal. 28, 1706–1716 (2018).

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