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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 Â鶹ÉçÇø, 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 â„“p-Improving Problem for Discrete Curves

Journal Article

Dendrinos, S., Hughes, K., & Vitturi, M. (2022)

Some Subcritical Estimates for the â„“p-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 Â鶹ÉçÇø, 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 â„“2 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 Â鶹ÉçÇø, 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...