7 papers
Space curve Fourier restriction for orthonormal functions
Jean-Claude Cuenin, Jacob Denson, Jonathan Hickman
We prove a variant of the Fourier restriction inequality for non-degenerate curves in the setting of orthonormal systems. To achieve this, we develop a Hilbert space approach…
Bourgain's proof of the circular maximal theorem: an exposition
Halit Yigit Gunduz, Jonathan Hickman
We present an exposition of Bourgain's celebrated 1986 proof of the circular maximal theorem.
Counting rational points near manifolds: a refined estimate, a conjecture and a variant
Jonathan Hickman, Rajula Srivastava, James Wright
Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of of intermediate dimension under a natu…
The Kakeya Conjecture: where does it come from and why is it important?
Jonathan Hickman
Roughly speaking, the Kakeya Conjecture asks to what extent lines which point in different directions can be packed together in a small space. In , the problem is relatively…
Counting integral points near space curves: a Fourier analytic approach
Jonathan Hickman, Rajula Srivastava
We establish upper and lower bounds for the number of integral points which lie within a neighbourhood of a smooth nondegenerate curve in for . These estima…
Improved bounds for the strong spherical maximal operator
Jonathan Hickman, Joshua Zahl
We study the mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this op…
Spherical maximal estimates via geometry
Jonathan Hickman, Ajša Jančar
We present a simple geometric approach to studying the boundedness properties of Stein's spherical maximal operator, which does not rely on the Fourier transform. Using this,…