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Improved bounds on number fields of small degree (2024)
Journal Article
Anderson, T. C., Gafni, A., Hughes, K., Lemke Oliver, R. J., Lowry-Duda, D., Thorne, F., Wang, J., & Zhang, R. (2024). Improved bounds on number fields of small degree. Discrete Analysis, 19, https://doi.org/10.19086/da.127737

We study the number of degree n number fields with discriminant bounded by X. In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for 6 ≤ n ≤ 94.

The Pointillist principle for variation operators and jump functions (2024)
Journal Article
Hughes, K. (online). The Pointillist principle for variation operators and jump functions. Revista de la Unión Matemática Argentina, https://doi.org/10.33044/revuma.4124

I extend the pointillist principles of Moon and Carrillo-de Guzmán to variational operators and jump functions. 1. The pointillist principle In [11], Moon observed that, for a sequence of sufficiently smooth convolution operators and any q ≥ 1, the w... Read More about The Pointillist principle for variation operators and jump functions.

A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem (2023)
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 precisely solution counting in older partial progress on Vinogradov's mean valu... Read More about A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem.

Discrete restriction estimates for forms in many variables (2023)
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 Tomas–Stein method and Magyar’s decomposition of the Fourier transform of... Read More about Discrete restriction estimates for forms in many variables.

Some Subcritical Estimates for the ℓp-Improving Problem for Discrete Curves (2022)
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 the number of solutions to certain special systems of diophantine equations,... Read More about Some Subcritical Estimates for the ℓp-Improving Problem for Discrete Curves.

On the inhomogeneous Vinogradov system (2022)
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 s<k(k+1)/2
than its homogeneous counterpart, provided that a_ℓ≠0 for some ℓ⩽k−1. Our methods use Vinogrado...

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.

Lp-improving for discrete spherical averages (2020)
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 discrete spherical averages and some of their generalizations. As an applic... Read More about Lp-improving for discrete spherical averages.

Lp→Lq bounds for spherical maximal operators (2020)
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 maximal function supt∈E|Atf|. A new feature is the dependence of the results o... Read More about Lp→Lq bounds for spherical maximal operators.