All Outputs (3)

On the ergodic Waring–Goldbach problem (2021)
Journal Article
Anderson, T. C., Cook, B., Hughes, K., & Kumchev, A. (2022). On the ergodic Waring–Goldbach problem. Journal of Functional Analysis, 282(5), Article 109334. https://doi.org/10.1016/j.jfa.2021.109334

We prove an asymptotic formula for the Fourier transform of the arithmetic surface measure associated to the Waring–Goldbach problem and provide several applications, including bounds for discrete spherical maximal functions along the primes and dist... Read More about On the ergodic Waring–Goldbach problem.

Discrete Restriction for (x,x3) and Related Topics (2021)
Journal Article
Hughes, K., & Wooley, T. D. (2022). Discrete Restriction for (x,x3) and Related Topics. International Mathematics Research Notices, 2022(20), 15612-15631. https://doi.org/10.1093/imrn/rnab113

In this short note we prove an ℓ2 to L10 estimate for the extension (aka adjoint restriction) operator associated to the discrete curve (X,X3). This is interesting, in part, because Demeter has shown that the corresponding putative decoupling inequal... Read More about Discrete Restriction for (x,x3) and Related Topics.

Bounds for Lacunary maximal functions given by Birch–Magyar averages (2021)
Journal Article
Cook, B., & Hughes, K. (2021). Bounds for Lacunary maximal functions given by Birch–Magyar averages. Transactions of the American Mathematical Society, 374(6), 3859-3879. https://doi.org/10.1090/tran/8152

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem di... Read More about Bounds for Lacunary maximal functions given by Birch–Magyar averages.