Kevin Hughes Jr
kevin hughes jr

Dr Kevin Hughes Jr

  

Biography

See my personal webpage for more information:
https://sites.google.com/site/khughesmath/home

My research areas harmonic analysis and analytic number theory.

10/2022 - Present: I am a lecturer (assistant professor) at the Edinburgh Napier University.

12/2015 - 09/2022: I was a research fellow (aka postdoc) at the University of Bristol in the analysis and number theory groups. My mentors were Trevor Wooley and John Mackay.

12/2012 - 12/2015: I was a Seggie Brown postdoctoral research fellow at The University of Edinburgh in the School of Mathematics. My mentors were Tony Carbery and Jim Wright.

In 2012, I completed my PhD at Princeton University, under the direction of Elias Stein, in `Arithmetic Analogues in Harmonic Analysis'.

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9 results

A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

Journal Article
Cook, B., Hughes, K., Li, Z. K., Mudgal, A., Robert, O., & Yung, P. (2024)
A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem. Mathematika, 70(1), Article e12231. https://doi.org/10.1112/mtk.12231
We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what prec...

Discrete restriction estimates for forms in many variables

Journal Article
Cook, B., Hughes, K., & Palsson, E. (2023)
Discrete restriction estimates for forms in many variables. Proceedings of the Edinburgh Mathematical Society, 66(4), 923–939. https://doi.org/10.1017/s0013091523000366
We prove discrete restriction estimates for a broad class of hypersurfaces arising in seminal work of Birch. To do so, we use a variant of Bourgain’s arithmetic version of the...

Some Subcritical Estimates for the ℓp-Improving Problem for Discrete Curves

Journal Article
Dendrinos, S., Hughes, K., & Vitturi, M. (2022)
Some Subcritical Estimates for the ℓp-Improving Problem for Discrete Curves. Journal of Fourier Analysis and Applications, 28(4), Article 69. https://doi.org/10.1007/s00041-022-09958-y
We apply Christ’s method of refinements to the ℓ^p-improving problem for discrete averages AN along polynomial curves in Z^d. Combined with certain elementary estimates for th...

On the inhomogeneous Vinogradov system

Journal Article
Brandes, J., & Hughes, K. (2022)
On the inhomogeneous Vinogradov system. Bulletin of the Australian Mathematical Society, 106(3), 396-403. https://doi.org/10.1017/s0004972722000284
We show that the system of equations ∑_{i=1}^{s} (x_i^j−y_i^j) = a_j (1⩽j⩽k) has appreciably fewer solutions in the subcritical range sthan its homogeneous counterpart, p...

On the ergodic Waring–Goldbach problem

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, includin...

Discrete Restriction for (x,x3) and Related Topics

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, b...

Bounds for Lacunary maximal functions given by Birch–Magyar averages

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 ...

Lp-improving for discrete spherical averages

Journal Article
Hughes, K. (2020)
Lp-improving for discrete spherical averages. Annales Henri Lebesgue, 3, 959-980. https://doi.org/10.5802/ahl.50
We initiate the theory of -improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove -improving estimates for the ...

Lp→Lq bounds for spherical maximal operators

Journal Article
Anderson, T., Hughes, K., Roos, J., & Seeger, A. (2021)
Lp→Lq bounds for spherical maximal operators. Mathematische Zeitschrift, 297(3-4), 1057-1074. https://doi.org/10.1007/s00209-020-02546-0
Let f∈Lp(Rd), d≥3, and let Atf(x) be the average of f over the sphere with radius t centered at x. For a subset E of [1, 2] we prove close to sharp Lp→Lq estimates for the max...

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