Cambridge Journal of Mathematics

Volume 7 (2019)

Number 3

Improved Fourier restriction estimates in higher dimensions

Pages: 219 – 282



Jonathan Hickman (Mathematical Institute, University of St Andrews, Fife, Scotland)

Keith M. Rogers (Instituto de Ciencias Matemáticas, Cantoblanco, Madrid, Spain)


We consider Guth’s approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.


Fourier transform, polynomial partitioning

2010 Mathematics Subject Classification


The first author is supported by the EPSRC standard grant EP/R015104/1 and the NSF grant DMS-1440140.

The second author is supported by the MINECO grants SEV-2015-0554 and MTM2017-85934-C3-1-P and the ERC grant 834728.

Received 4 March 2019

Published 11 September 2019