Scholars@Duke
Lillian Beatrix Pierce

Lillian Beatrix Pierce

Anne Firor Scott Distinguished Professor of Mathematics
Mathematics
pierce@math.duke.edu
Duke Box 90320, Room 117 Physics Building, Durham, NC 27708
Duke Box 90320 120 Science Drive,, Durham, NC 27708

Scholarly Works

Genuine and strongly genuine polynomials: With an application to the persistence of Galois groups under specialization

Preprint · July 2, 2026 We develop the theory of strongly $n$-genuine polynomials $F(Y,X_1,\ldots,X_n)$, which have the property that the number of specializations $F(Y,X_1,\mathbf{x}')$ with $\mathbf{x}'=(x_2,\ldots,x_n) \in \mathbb{Z}^{n-1}$ (respectively $\mathbf{x}' \in \math ... Link to item Cite

A guide to Tauberian theorems for arithmetic applications

Preprint · April 22, 2025 A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging from well-known ... Link to item Cite

GENERALISED QUADRATIC FORMS OVER TOTALLY REAL NUMBER FIELDS

Journal article Journal of the Institute of Mathematics of Jussieu · November 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 text Cite

On Polynomial Carleson Operators Along Quadratic Hypersurfaces

Journal article Journal 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 text Cite

A NEW TYPE OF SUPERORTHOGONALITY

Journal article Proceedings 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 o ... Full text Cite

Application of a polynomial sieve: Beyond separation of variables

Journal article Algebra and Number Theory · January 1, 2024 Let a polynomial be given. The square sieve can provide an upper bound for the number of integral x ∊ [−B, B]n such that f(x) is a perfect square. Recently this has been generalized substantially: First to a power sieve, counting x ∊ [−B, B] Full text Cite

Generalizations of the Schrödinger maximal operator: building arithmetic counterexamples

Journal article Journal 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. converg ... Full text Cite

Counterexamples for High-Degree Generalizations of the Schrödinger Maximal Operator

Journal article International 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 text Cite

A new type of superorthogonality

Preprint · December 17, 2022 Link to item Cite

Geometric generalizations of the square sieve, with an application to cyclic covers

Journal article Mathematika: 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 item Cite

On the strict majorant property in arbitrary dimensions

Journal article Quarterly 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 item Cite

Reversing a Philosophy: From Counting to Square Functions and Decoupling

Journal article Journal 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 text Cite

On Superorthogonality

Journal article Journal 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 supero ... Full text Cite

Elias M. Stein (1931–2018)

Journal article Notices of the American Mathematical Society · April 2021 Full text Cite

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

Journal article Mathematical 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 item Cite

Burgess bounds for short character sums evaluated at forms II: the mixed case

Journal article Rivista 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

ON BOURGAIN’S COUNTEREXAMPLE for the SCHRÖDINGER MAXIMAL FUNCTION

Journal article Quarterly 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 text Cite

Analysis and applications: The mathematical work of Elias Stein

Journal article Bulletin of the American Mathematical Society · March 3, 2020 This article discusses some of Elias M. Stein’s seminal contributions to analysis. ... Full text Cite

An effective Chebotarev density theorem for families of number fields, with an application to $\ell$-torsion in class groups

Journal article Inventiones 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 ... Open Access Link to item Cite

On matrix rearrangement inequalities

Journal article Proceedings 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 ... Full text Cite

Burgess bounds for short character sums evaluated at forms

Journal article Algebra 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 ... Full text Cite

Burgess bounds for short character sums evaluated at forms II: the mixed case

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 item Cite

Séminaire Bourbaki Volume 2016/2017 Exposés 1120-1135

Journal article Asterisque · 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 ... Full text Cite

A polynomial Carleson operator along the paraboloid

Journal article Revista Matemática Iberoamericana · 2018 Link to item Cite

Endpoint Sobolev and BV continuity for maximal operators

Journal article Journal of Functional Analysis · November 2017 Full text Cite

The Vinogradov Mean Value Theorem [after Wooley, and Bourgain, Demeter and Guth]

Journal article AstÉrisque · 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. ... Link to item Cite

Simultaneous integer values of pairs of quadratic forms

Journal article Journal 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 ... Full text Open Access Cite

On ℓ-torsion in class groups of number fields

Journal article Algebra and Number Theory · 2017 © 2017 Mathematical Sciences Publishers. For each integer ℓ ≥ 1, we prove an unconditional upper bound on the size of the ℓ-torsion subgroup of the class group, which holds for all but a zerodensity set of field extensions of Q of degree d, for any fixed d ... Full text Cite

Polynomial Carleson operators along monomial curves in the plane

Journal article Journal 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 ... Link to item Cite

Burgess bounds for multi-dimensional short mixed character sums

Journal article Journal 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 ... Full text Cite

Representations of integers by systems of three quadratic forms

Journal article Proceedings 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 ... Link to item Cite

Burgess bounds for short mixed character sums

Journal article Journal of the London Mathematical Society · 2015 © 2015 London Mathematical Society.This paper proves non-trivial bounds for short mixed character sums by introducing estimates for Vinogradov's mean value theorem into a version of the Burgess method. ... Full text Cite

Corrigendum to “on a discrete version of Tanaka’s theorem for maximal functions”

Journal article Proceedings of the American Mathematical Society · 2015 © 2015 American Mathematical Society.In this note we present a brief fix for an oversight in the proof of Lemma 3(iii) in a 2012 paper by Bober, Carneiro, Hughes and Pierce. ... Full text Cite

Lower bounds for the truncated Hilbert transform

Journal article arXiv: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 ... Link to item Cite

On a discrete version of Tanaka's theorem for maximal functions

Journal article Proceedings 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 ... Full text Link to item Cite

A note on discrete fractional integral operators on the heisenberg group

Journal article International Mathematics Research Notices · 2012 We consider the discrete analog of a fractional integral operator on the Heisenberg group, for which we are able to prove nearly sharp results by means of a simple argument of a combinatorial nature. © 2011 The Author(s). Published by Oxford University Pre ... Full text Cite

Counting rational points on smooth cyclic covers

Journal article Journal 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 ... Full text Cite

Discrete fractional radon transforms and quadratic forms

Journal article Duke 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 ... Full text Cite

On discrete fractional integral operators and mean values of Weyl sums

Journal article Bulletin 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 ... Full text Cite

A note on twisted discrete singular Radon transforms

Journal article Mathematical 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 ... Cite

A bound for the 3-part of class numbers of quadratic fields by means of the square sieve

Journal article Forum 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 ... Full text Cite

The 3-part of class numbers of quadratic fields

Journal article Journal 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 ... Full text Cite