Journal ArticleJournal of Geometric Analysis · October 1, 2024
We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by (y,Q(y))⊆Rn+1, for an arbitrary non-degenerate quadratic form Q, admits an a priori bound on Lp for all 1
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Journal ArticleProceedings of the American Mathematical Society · February 1, 2024
We provide a simple criterion on a family of functions that implies a square function estimate on Lp for every even integer p ≥ 2. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type o ...
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Journal ArticleJournal of the Institute of Mathematics of Jussieu · January 1, 2024
We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Li ...
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Journal ArticleJournal d'Analyse Mathematique · December 1, 2023
Let TtP2f(x) denote the solution to the linear Schrödinger equation at time t, with initial value function f, where P 2(ξ) = ∣ξ∣2. In 1980, Carleson asked for the minimal regularity of f that is required for the pointwise a.e. convergence of TtP2f(x) to f( ...
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Journal ArticleInternational Mathematics Research Notices · May 1, 2023
In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (u ...
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Journal ArticleMathematika: a journal of pure and applied mathematics · 2022
We formulate a general problem: given projective schemes $\mathbb{Y}$ and
$\mathbb{X}$ over a global field $K$ and a $K$-morphism $\eta$ from
$\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in
$\mathbb{X}(K)$ of height at most $B$ have a pre ...
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Journal ArticleQuarterly Journal of Mathematics · 2022
In this work we study $d$-dimensional majorant properties. We prove that a
set of frequencies in ${\mathbb Z}^d$ satisfies the strict majorant property on
$L^p([0,1]^d)$ for all $p> 0$ if and only if the set is affinely independent.
We further construct th ...
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Journal ArticleJournal of Geometric Analysis · July 1, 2021
Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations thi ...
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Journal ArticleJournal of Geometric Analysis · July 1, 2021
In this survey, we explore how superorthogonality amongst functions in a sequence f1, f2, f3, … results in direct or converse inequalities for an associated square function. We distinguish between three main types of superorthogonality, which we demonstrat ...
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Journal ArticleMathematical Research Letters · 2021
It is conjectured that within the class group of any number field, for every
integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an
appropriate sense, relative to the discriminant of the field). In nearly all
settings, the full strength of ...
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Journal ArticleRivista di Matematica della Universita di Parma · January 1, 2021
This work proves a Burgess bound for short mixed character sums in n dimensions. The non-principal multiplicative character of prime conductor q may be evaluated at any "admissible" form, and the additive character may be evaluated at any real-valued polyn ...
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Journal ArticleQuarterly Journal of Mathematics · December 1, 2020
This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schrödinger equation, for initial data in a Sobolev space. This counterexample combines ...
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Journal ArticleJournal of Geometric Analysis · October 1, 2024
We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by (y,Q(y))⊆Rn+1, for an arbitrary non-degenerate quadratic form Q, admits an a priori bound on Lp for all 1
Full textCite
Journal ArticleProceedings of the American Mathematical Society · February 1, 2024
We provide a simple criterion on a family of functions that implies a square function estimate on Lp for every even integer p ≥ 2. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type o ...
Full textCite
Journal ArticleJournal of the Institute of Mathematics of Jussieu · January 1, 2024
We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Li ...
Full textCite
Journal ArticleJournal d'Analyse Mathematique · December 1, 2023
Let TtP2f(x) denote the solution to the linear Schrödinger equation at time t, with initial value function f, where P 2(ξ) = ∣ξ∣2. In 1980, Carleson asked for the minimal regularity of f that is required for the pointwise a.e. convergence of TtP2f(x) to f( ...
Full textCite
Journal ArticleInternational Mathematics Research Notices · May 1, 2023
In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (u ...
Full textCite
Journal ArticleMathematika: a journal of pure and applied mathematics · 2022
We formulate a general problem: given projective schemes $\mathbb{Y}$ and
$\mathbb{X}$ over a global field $K$ and a $K$-morphism $\eta$ from
$\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in
$\mathbb{X}(K)$ of height at most $B$ have a pre ...
Link to itemCite
Journal ArticleQuarterly Journal of Mathematics · 2022
In this work we study $d$-dimensional majorant properties. We prove that a
set of frequencies in ${\mathbb Z}^d$ satisfies the strict majorant property on
$L^p([0,1]^d)$ for all $p> 0$ if and only if the set is affinely independent.
We further construct th ...
Link to itemCite
Journal ArticleJournal of Geometric Analysis · July 1, 2021
Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations thi ...
Full textCite
Journal ArticleJournal of Geometric Analysis · July 1, 2021
In this survey, we explore how superorthogonality amongst functions in a sequence f1, f2, f3, … results in direct or converse inequalities for an associated square function. We distinguish between three main types of superorthogonality, which we demonstrat ...
Full textCite
Journal ArticleMathematical Research Letters · 2021
It is conjectured that within the class group of any number field, for every
integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an
appropriate sense, relative to the discriminant of the field). In nearly all
settings, the full strength of ...
Link to itemCite
Journal ArticleRivista di Matematica della Universita di Parma · January 1, 2021
This work proves a Burgess bound for short mixed character sums in n dimensions. The non-principal multiplicative character of prime conductor q may be evaluated at any "admissible" form, and the additive character may be evaluated at any real-valued polyn ...
Cite
Journal ArticleQuarterly Journal of Mathematics · December 1, 2020
This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schrödinger equation, for initial data in a Sobolev space. This counterexample combines ...
Full textCite
Journal ArticleInventiones Mathematicae · 2020
An effective Chebotarev density theorem for a fixed normal extension
$L/\mathbb{Q}$ provides an asymptotic, with an explicit error term, for the
number of primes of bounded size with a prescribed splitting type in $L$. In
many applications one is most inte ...
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Journal ArticleProceedings of the American Mathematical Society · January 1, 2020
Given two symmetric and positive semidefinite square matrices A,B, is it true that any matrix given as the product of m copies of A and n copies of B in a particular sequence must be dominated in the spectral norm by the ordered matrix product AmBn? For ex ...
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Journal ArticleAlgebra and Number Theory · January 1, 2020
We establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of “admissible” forms. This n-dimensional Burgess bound is nontrivi ...
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Journal Article · 2020
This work proves a Burgess bound for short mixed character sums in $n$
dimensions. The non-principal multiplicative character of prime conductor $q$
may be evaluated at any "admissible" form, and the additive character may be
evaluated at any real-valued p ...
Link to itemCite
Journal ArticleAsterisque · January 1, 2019
This 69th volume of the Bourbaki Seminar contains the texts of the fifteen survey lectures done during the year 2016/2017. Topics addressed covered Langlands correspondence, NIP property in model theory, Navier–Stokes equation, algebraic and complex analyt ...
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Journal ArticleAstÉrisque · July 4, 2017
This is the expository essay that accompanies my Bourbaki Seminar on 17 June
2017 on the landmark proof of the Vinogradov Mean Value Theorem, and the two
approaches developed in the work of Wooley and of Bourgain, Demeter and Guth. ...
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Journal ArticleJournal fur die Reine und Angewandte Mathematik · June 1, 2017
We prove that a pair of integral quadratic forms in five or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In par ...
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Journal ArticleJournal of Geometric Analysis · 2017
We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing sto ...
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Journal ArticleJournal of Number Theory · June 1, 2016
This paper proves Burgess bounds for short mixed character sums in multi-dimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial f, and a product of multiplicative Dirichlet ...
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Journal ArticleProceedings of the London Mathematical Society · 2016
It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers $(n_1,\ldots,n_R)$ by a system of quadratic forms $Q_1,\ldots, Q_R$ in $k$ variables, as long as $k$ is sufficiently large; reduc ...
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Journal ArticlearXiv:1311.6845 [math] · November 26, 2013
Given two intervals $I, J \subset \mathbb{R}$, we ask whether it is possible to reconstruct a real-valued function $f \in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. When neither interval is fully contained in the other, this problem has a uniq ...
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Journal ArticleProceedings of the American Mathematical Society · May 1, 2012
In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered maximal operator $\widetilde{M} $ we prove that ...
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Journal ArticleJournal of Number Theory · 2012
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space Pn-1. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n≥ 10, and surpass it for ...
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Journal ArticleDuke Mathematical Journal · 2012
We consider discrete analogues of fractional Radon transforms involving integration over paraboloids defined by positive definite quadratic forms. We prove sharp results for this class of discrete operators in all dimensions, providing necessary and suffic ...
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Journal ArticleBulletin of the London Mathematical Society · 2011
In this paper, we prove new ℓp→ℓq bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit ...
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Journal ArticleMathematical Research Letters · 2010
In this paper we consider three types of discrete operators stemming from singular Radon transforms. We first extend an ℓp result for translation invariant discrete singular Radon transforms to a class of twisted operators including an additional oscillato ...
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Journal ArticleForum Mathematicum · 2006
We prove a nontrivial bound of O(D27/56+ε) for the 3-part of the class number of a quadratic field (√D) by using a variant of the square sieve and the q-analogue of van der Corput's method to count the number of squares of the form 4x3 - dz2 for a square-f ...
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Journal ArticleJournal of the London Mathematical Society · 2005
It is proved that the 3-part of the class number of a quadratic field ℚ(√D) is O(|D|55/112+ε) in general and O(|D| 5/12+ε) if |D| has a divisor of size |D|5/6. These bounds follow as results of nontrivial estimates for the number of solutions to the congru ...
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