51 citations · 138 across the 17 of their papers we have counts for
21 papers
Reinforcing a Philosophy: Littlewood--Paley theory for the moment curve over generic local fields
Kevin Hughes
Using the Girard--Newton formulae, I give a simple proof of isotropic square function estimates for extension operators along the moment curve in generic local fields. Using Bezout…
A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem
Brian Cook, Kevin Hughes, Zane Kun Li +3
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…
Improved bounds on number fields of small degree
Theresa C. Anderson, Ayla Gafni, Kevin Hughes +5
We study the number of degree number fields with discriminant bounded by . In this article, we improve an upper bound due to Schmidt on the number of such fields that was pr…
Subcritical paucity and -improving estimates for finite-type polynomial curves
Kevin Hughes
I prove new subcritical bounds for the -improving problem along restricted subsets of a degenerate curve. The key input is a new paucity estimate for associated inhomogeneo…
Reinforcing a Philosophy: A counting approach to square functions over local fields
Kirsti D. Biggs, Julia Brandes, Kevin Hughes
In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of th…
Some subcritical estimates for the -improving problem for discrete curves
Spyridon Dendrinos, Kevin Hughes, Marco Vitturi
We apply Christ's method of refinements to the -improving problem for discrete averages along polynomial curves in . Combined with certain ele…
Supercritical discrete restriction estimates for forms in many variables
Brian Cook, Kevin Hughes, Eyvindur Palsson
We prove discrete restriction estimates for a broad class of hypersurfaces and varieties of intermediate codimension. For our result about hypersurfaces, we use Bourgain's arithmet…
Discrete restriction for and related topics
Kevin Hughes, Trevor D. Wooley
Defining the truncated extension operator for a sequence with by putting \[ E{a}(α,β):=\sum_{|n|\leq N}a(n) e(αn^3 + βn), \] we obtain the conjecture…
Bounds for Lacunary maximal functions given by Birch--Magyar averages
Brian Cook, Kevin Hughes
We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equat…
-improving for discrete spherical averages
Kevin Hughes
We initiate the theory of -improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove -improving estimat…
Improved -Boundedness for Integral -Spherical Maximal Functions
Theresa C. Anderson, Brian Cook, Kevin Hughes +1
We improve the range of -boundedness of the integral -spherical maximal functions introduced by Magyar. The previously best known bounds for the full -sp…
On the ergodic Waring--Goldbach problem
Theresa C. Anderson, Brian Cook, Kevin Hughes +1
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 bo…
On restriction estimates for discrete quadratic surfaces
Kevin Henriot, Kevin Hughes
We obtain truncated restriction estimates of an unexpected form for discrete surfaces \begin{align} S = \{\, ( n_1 , \dots , n_d , R( n_1 , \dots, n_d ) ) \,,\, n_i \in [-N,N] \cap…
Discrete restriction estimates of epsilon-removal type for kth-powers and k-paraboloids
Kevin Henriot, Kevin Hughes
We obtain restriction estimates of -removal type for the set of -th powers of integers, and for discrete -dimensional surfaces of the form \[ \{ (n_1,\dots,n_d,n_1^k + \do…
The Pointillist principle for variation operators and jump functions
Kevin Hughes
We extend the pointillist principles of Moon and Carrillo--de Guzmán to variational operators and jump functions.
The discrete spherical averages over a family of sparse sequences
Kevin Hughes
We initiate the study of the -boundedness of the arithmetic spherical maximal function over sparse sequences. We state a folklore conjecture for lacunary sequ…
Restricted weak-type endpoint estimates for discrete k-spherical maximal functions
Kevin Hughes
In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the l^p(Z…
The Maximal Function and Square Function Control the Variation: An Elementary Proof
Kevin Hughes, Ben Krause, Bartosz Trojan
In this note we prove the following good- inequality, for , all , \[ ν\big\{ V_r(f) > 3 λ; \mathcal{M}(f) \leq δλ\big\} \leq 4 ν\{s(f…
Problems and Results related to Waring's problem: Maximal functions and ergodic averages
Kevin Hughes
We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of \cite{Magyar_dyadic}, \cite{Magyar_ergodic} an…
On the endpoint regularity of discrete maximal operators
Emanuel Carneiro, Kevin Hughes
Given a discrete function we consider the maximal operator …
On a discrete version of Tanaka's theorem for maximal functions
Jonathan Bober, Emanuel Carneiro, Kevin Hughes +1
In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the Hardy-Littlewood maximal operator in dimension , both in the non-centered and centered cases. F…