%% Papers Published
@article{fds375267,
Author = {Gressman, PT and Pierce, LB and Roos, J and Yung,
PL},
Title = {A NEW TYPE OF SUPERORTHOGONALITY},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {152},
Number = {2},
Pages = {665-675},
Year = {2024},
Month = {February},
url = {http://dx.doi.org/10.1090/proc/16631},
Abstract = {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 of
superorthogonality that is currently known.},
Doi = {10.1090/proc/16631},
Key = {fds375267}
}
@article{fds375268,
Author = {Chu, R and Pierce, LB},
Title = {Generalizations of the Schrödinger maximal operator:
building arithmetic counterexamples},
Journal = {Journal d'Analyse Mathematique},
Volume = {151},
Number = {1},
Pages = {59-114},
Year = {2023},
Month = {December},
url = {http://dx.doi.org/10.1007/s11854-023-0335-7},
Abstract = {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(x) as t → 0. This was
recently resolved by work of Bourgain, and Du and Zhang.
This paper considers more general dispersive equations, and
constructs counterexamples to pointwise a.e. convergence for
a new class of real polynomial symbols P of arbitrary
degree, motivated by a broad question: what occurs for
symbols lying in a generic class? We construct the
counterexamples using number-theoretic methods, in
particular the Weil bound for exponential sums, and the
theory of Dwork-regular forms. This is the first case in
which counterexamples are constructed for indecomposable
forms, moving beyond special regimes where P has some
diagonal structure.},
Doi = {10.1007/s11854-023-0335-7},
Key = {fds375268}
}
@article{fds374482,
Author = {Pierce, L and Chu, R},
Title = {Generalizations of the Schrödinger maximal operator:
building arithmetic counterexamples},
Booktitle = {https://arxiv.org/abs/2309.05872},
Year = {2023},
Month = {November},
Key = {fds374482}
}
@article{fds370839,
Author = {An, C and Chu, R and Pierce, LB},
Title = {Counterexamples for High-Degree Generalizations of the
Schrödinger Maximal Operator},
Journal = {International Mathematics Research Notices},
Volume = {2023},
Number = {10},
Pages = {8371-8418},
Publisher = {Oxford University Press (OUP)},
Year = {2023},
Month = {May},
url = {http://dx.doi.org/10.1093/imrn/rnac088},
Abstract = {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 (up to the endpoint) by Bourgain,
whose counterexample construction for the Schrödinger
maximal operator proved a necessary condition on the
regularity, and Du and Zhang, who proved a sufficient
condition. Analogues of Carleson's question remain open for
many other dispersive partial differential equations. We
develop a flexible new method to approach such problems and
prove that for any integer, if a degree generalization of
the Schrödinger maximal operator is bounded from to, then
In dimensions, for every degree, this is the first result
that exceeds a long-standing barrier at. Our methods are
number-theoretic, and in particular apply the Weil bound, a
consequence of the truth of the Riemann Hypothesis over
finite fields.},
Doi = {10.1093/imrn/rnac088},
Key = {fds370839}
}
@article{fds369111,
Author = {Browning, T and Pierce, LB and Schindler, D},
Title = {Generalised quadratic forms over totally real number
fields},
Year = {2022},
Month = {December},
Key = {fds369111}
}
@article{fds368513,
Author = {Gressman, PT and Pierce, LB and Roos, J and Yung,
P-L},
Title = {A new type of superorthogonality},
Year = {2022},
Month = {December},
Key = {fds368513}
}
@article{fds368139,
Author = {Anderson, TC and Maldague, D and Pierce, L},
Title = {On Polynomial Carleson operators along quadratic
hypersurfaces},
Year = {2022},
Month = {November},
Key = {fds368139}
}
@article{fds365843,
Author = {Bonolis, D and Pierce, LB},
Title = {Application of a polynomial sieve: beyond separation of
variables},
Year = {2022},
Month = {September},
Abstract = {Let a polynomial $f \in \mathbb{Z}[X_1,\ldots,X_n]$ be
given. The square sieve can provide an upper bound for the
number of integral $\mathbf{x} \in [-B,B]^n$ such that
$f(\mathbf{x})$ is a perfect square. Recently this has been
generalized substantially: first to a power sieve, counting
$\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=y^r$ is
solvable for $y \in \mathbb{Z}$; then to a polynomial sieve,
counting $\mathbf{x} \in [-B,B]^n$ for which
$f(\mathbf{x})=g(y)$ is solvable, for a given polynomial
$g$. Formally, a polynomial sieve lemma can encompass the
more general problem of counting $\mathbf{x} \in [-B,B]^n$
for which $F(y,\mathbf{x})=0$ is solvable, for a given
polynomial $F$. Previous applications, however, have only
succeeded in the case that $F(y,\mathbf{x})$ exhibits
separation of variables, that is, $F(y,\mathbf{x})$ takes
the form $f(\mathbf{x}) - g(y)$. In the present work, we
present the first application of a polynomial sieve to count
$\mathbf{x} \in [-B,B]^n$ such that $F(y,\mathbf{x})=0$ is
solvable, in a case for which $F$ does not exhibit
separation of variables. Consequently, we obtain a new
result toward a question of Serre, pertaining to counting
points in thin sets.},
Key = {fds365843}
}
@article{fds360132,
Author = {Bucur, A and Cojocaru, AC and Lalín, MN and Pierce,
LB},
Title = {Geometric generalizations of the square sieve, with an
application to cyclic covers},
Journal = {Mathematika: a journal of pure and applied
mathematics},
Publisher = {Wiley},
Year = {2022},
Abstract = {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-image under $\eta$ in
$\mathbb{Y}(K)$? This problem is inspired by a well-known
conjecture of Serre on quantitative upper bounds for the
number of points of bounded height on an irreducible
projective variety defined over a number field. We give a
non-trivial answer to the general problem when
$K=\mathbb{F}_q(T)$ and $\mathbb{Y}$ is a prime degree
cyclic cover of $\mathbb{X}=\mathbb{P}_{K}^n$. Our tool is a
new geometric sieve, which generalizes the polynomial sieve
to a geometric setting over global function
fields.},
Key = {fds360132}
}
@article{fds360131,
Author = {Gressman, PT and Guo, S and Pierce, LB and Roos, J and Yung,
P-L},
Title = {On the strict majorant property in arbitrary
dimensions},
Journal = {Quarterly Journal of Mathematics},
Publisher = {Oxford University Press},
Year = {2022},
Abstract = {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 three types of violations of the strict
majorant property. Any set of at least $d+2$ frequencies in
${\mathbb Z}^d$ violates the strict majorant property on
$L^p([0,1]^d)$ for an open interval of $p \not\in 2 {\mathbb
N}$ of length 2. Any infinite set of frequencies in
${\mathbb Z}^d$ violates the strict majorant property on
$L^p([0,1]^d)$ for an infinite sequence of open intervals of
$p \not\in 2 {\mathbb N}$ of length $2$. Finally, given any
$p>0$ with $p \not\in 2{\mathbb N}$, we exhibit a set of
$d+2$ frequencies on the moment curve in ${\mathbb R}^d$
that violate the strict majorant property on
$L^p([0,1]^d).$},
Key = {fds360131}
}
@article{fds355712,
Author = {Pierce, LB},
Title = {On Superorthogonality},
Journal = {Journal of Geometric Analysis},
Volume = {31},
Number = {7},
Pages = {7096-7183},
Year = {2021},
Month = {July},
url = {http://dx.doi.org/10.1007/s12220-021-00606-3},
Abstract = {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 demonstrate arise in a wide array of settings in
harmonic analysis and number theory. This perspective gives
clean proofs of central results, and unifies topics
including Khintchine’s inequality, Walsh–Paley series,
discrete operators, decoupling, counting solutions to
systems of Diophantine equations, multicorrelation of trace
functions, and the Burgess bound for short character
sums.},
Doi = {10.1007/s12220-021-00606-3},
Key = {fds355712}
}
@article{fds355326,
Author = {Gressman, PT and Guo, S and Pierce, LB and Roos, J and Yung,
PL},
Title = {Reversing a Philosophy: From Counting to Square Functions
and Decoupling},
Journal = {Journal of Geometric Analysis},
Volume = {31},
Number = {7},
Pages = {7075-7095},
Year = {2021},
Month = {July},
url = {http://dx.doi.org/10.1007/s12220-020-00593-x},
Abstract = {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 this implication can also be
reversed. As a first example, we observe that a count for
the number of integral solutions to a system of Diophantine
equations implies a discrete decoupling inequality. Second,
in our main result we prove an L2n square function estimate
(which implies a corresponding decoupling estimate) for the
extension operator associated to a non-degenerate curve in
Rn. The proof is via a combinatorial argument that builds on
the idea that if γ is a non-degenerate curve in Rn, then as
long as x1, … , x2n are chosen from a sufficiently
well-separated set, then γ(x1) + ⋯ + γ(xn) = γ(xn+1) +
⋯ + γ(x2n) essentially only admits solutions in which x1,
… , xn is a permutation of xn+1, … ,
x2n.},
Doi = {10.1007/s12220-020-00593-x},
Key = {fds355326}
}
@article{fds356099,
Author = {Pierce, LB and Beckner, W and Dafni, G and Fefferman, C and Ionescu, A and Kearn, V and Kenig, CE and Knapp, AW and Krantz, SG and Lanzani, L and Nagel, A and Phong, DH and Ricci, F and Rothschild, L and Shakarchi, R and Sogge, C and Stein, J and Stein, K and Tao, T and Wainger, S and Widom,
H},
Title = {Elias M. Stein (1931–2018)},
Journal = {Notices of the American Mathematical Society},
Volume = {68},
Number = {04},
Pages = {1-1},
Publisher = {American Mathematical Society (AMS)},
Year = {2021},
Month = {April},
url = {http://dx.doi.org/10.1090/noti2264},
Doi = {10.1090/noti2264},
Key = {fds356099}
}
@article{fds360133,
Author = {Pierce, L},
Title = {Counting problems: class groups, primes, and number fields,
ICM 2022 Proceedings (accepted, in press)},
Year = {2021},
Key = {fds360133}
}
@article{fds356100,
Author = {Pierce, LB and Turnage-Butterbaugh, CL and Wood,
MM},
Title = {On a conjecture for $\ell$-torsion in class groups of number
fields: from the perspective of moments},
Journal = {Mathematical Research Letters},
Volume = {28},
Number = {2},
Pages = {575-621},
Publisher = {International Press},
Year = {2021},
Abstract = {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 this conjecture remains open, and even
partial progress is limited. Significant recent progress
toward average versions of the $\ell$-torsion conjecture has
crucially relied on counts for number fields, raising
interest in how these two types of question relate. In this
paper we make explicit the quantitative relationships
between the $\ell$-torsion conjecture and other well-known
conjectures: the Cohen-Lenstra heuristics, counts for number
fields of fixed discriminant, counts for number fields of
bounded discriminant (or related invariants), and counts for
elliptic curves with fixed conductor. All of these
considerations reinforce that we expect the $\ell$-torsion
conjecture is true, despite limited progress toward it. Our
perspective focuses on the relation between pointwise
bounds, averages, and higher moments, and demonstrates the
broad utility of the "method of moments."},
Key = {fds356100}
}
@article{fds354950,
Author = {Pierce, LB},
Title = {ON BOURGAIN’S COUNTEREXAMPLE for the SCHRÖDINGER MAXIMAL
FUNCTION},
Journal = {Quarterly Journal of Mathematics},
Volume = {71},
Number = {4},
Pages = {1309-1344},
Publisher = {Oxford University Press (OUP)},
Year = {2020},
Month = {December},
url = {http://dx.doi.org/10.1093/qmath/haaa032},
Abstract = {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 ideas from analysis and number
theory, and the present paper demonstrates how to build such
counterexamples from first principles, and then optimize
them.},
Doi = {10.1093/qmath/haaa032},
Key = {fds354950}
}
@article{fds356102,
Author = {Fefferman, C and Ionescu, A and Tao, T and Wainger,
S},
Title = {Analysis and applications: The mathematical work of Elias
Stein},
Journal = {Bulletin of the American Mathematical Society},
Volume = {57},
Number = {4},
Pages = {523-594},
Publisher = {American Mathematical Society (AMS)},
Year = {2020},
Month = {March},
url = {http://dx.doi.org/10.1090/bull/1691},
Doi = {10.1090/bull/1691},
Key = {fds356102}
}
@article{fds349180,
Author = {Alaifari, R and Cheng, X and Pierce, LB and Steinerberger,
S},
Title = {On matrix rearrangement inequalities},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {148},
Number = {5},
Pages = {1835-1848},
Publisher = {American Mathematical Society (AMS)},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.1090/proc/14831},
Abstract = {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 example, is ∥ AABAABABB
∥ ≤ ∥AAAAABBBB∥? Drury [Electron J. Linear Algebra
18 (2009), pp. 13 20] has characterized precisely which
disordered words have the property that an inequality of
this type holds for all matrices A,B. However, the
1-parameter family of counterexamples Drury constructs for
these characterizations is comprised of 3×3 matrices, and
thus as stated the characterization applies only for N × N
matrices with N ≤ 3. In contrast, we prove that for 2 × 2
matrices, the general rearrangement inequality holds for all
disordered words. We also show that for larger N ×N
matrices, the general rearrangement inequality holds for all
disordered words for most A,B (in a sense of full measure)
that are sufficiently small perturbations of the
identity.},
Doi = {10.1090/proc/14831},
Key = {fds349180}
}
@article{fds352548,
Author = {Pierce, LB and Xu, J},
Title = {Burgess bounds for short character sums evaluated at
forms},
Journal = {Algebra and Number Theory},
Volume = {14},
Number = {7},
Pages = {1911-1951},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.2140/ant.2020.14.1911},
Abstract = {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 nontrivial for sums over
boxes of sidelength at least qβ, with β > 1/2 − 1/(2(n +
1)). This is the first Burgess bound that applies in all
dimensions to generic forms of arbitrary degree. Our
approach capitalizes on a recent stratification result for
complete multiplicative character sums evaluated at rational
functions, due to the second author.},
Doi = {10.2140/ant.2020.14.1911},
Key = {fds352548}
}
@article{fds330203,
Author = {Pierce, LB and Turnage-Butterbaugh, CL and Wood,
MM},
Title = {An effective Chebotarev density theorem for families of
number fields, with an application to $\ell$-torsion in
class groups},
Journal = {Inventiones Mathematicae},
Volume = {219},
Number = {2},
Pages = {707-778},
Publisher = {Springer},
Year = {2020},
url = {http://dx.doi.org/10.1007/s00222-019-00915-z},
Abstract = {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 interested in the case where the
primes are small (with respect to the absolute discriminant
of $L$); this is well-known to be closely related to the
Generalized Riemann Hypothesis for the Dedekind zeta
function of $L$. In this work we prove a new effective
Chebotarev density theorem, independent of GRH, that
improves the previously known unconditional error term and
allows primes to be taken quite small (certainly as small as
an arbitrarily small power of the discriminant of $L$); this
theorem holds for the Galois closures of "almost all" number
fields that lie in an appropriate family of field
extensions. Such a family has fixed degree, fixed Galois
group of the Galois closure, and in certain cases a
ramification restriction on all tamely ramified primes in
each field; examples include totally ramified cyclic fields,
degree $n$ $S_n$-fields with square-free discriminant, and
degree $n$ $A_n$-fields. In all cases, our work is
independent of GRH; in some cases we assume the strong Artin
conjecture or hypotheses on counting number fields. The new
effective Chebotarev theorem is expected to have many
applications, of which we demonstrate two. First we prove
(for all integers $\ell \geq 1$) nontrivial bounds for
$\ell$-torsion in the class groups of "almost all" fields in
the families of fields we consider. This provides the first
nontrivial upper bounds for $\ell$-torsion, for all integers
$\ell \geq 1$, applicable to infinite families of fields of
arbitrarily large degree. Second, in answer to a question of
Ruppert, we prove that within each family, "almost all"
fields have a small generator.},
Doi = {10.1007/s00222-019-00915-z},
Key = {fds330203}
}
@article{fds356103,
Author = {Pierce, LB},
Title = {Burgess bounds for short character sums evaluated at forms
II: the mixed case},
Volume = {12},
Number = {1},
Pages = {151-179},
Year = {2020},
Abstract = {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 polynomial. The resulting upper
bound for the mixed character sum is nontrivial when the
length of the sum is at least $q^{\beta}$ with $\beta> 1/2 -
1/(2(n+1))$ in each coordinate. This work capitalizes on the
recent stratification of multiplicative character sums due
to Xu, and the resolution of the Vinogradov Mean Value
Theorem in arbitrary dimensions.},
Key = {fds356103}
}
@article{fds347383,
Author = {Chruściel, PT and De Mesmay and A and Păun, M and Peyre, E and Barthe, F and Helfgott, HA and Kontsevich, M and Villani, C and Guillermou, S and Hernandez, D and Ma, X and Massot, P and Bergeron, N and Oesterlé, J and Pierce, LB and Rousset, F},
Title = {Séminaire Bourbaki Volume 2016/2017 Exposés
1120-1135},
Journal = {Asterisque},
Volume = {407},
Pages = {1-602},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.24033/ast.1057},
Abstract = {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 analytic
geometry, algorithmic and geometric questions in knot
theory, analytic number theory formal moduli problems,
general relativity, sofic entropy, sphere packings,
subriemannian geometry.},
Doi = {10.24033/ast.1057},
Key = {fds347383}
}
@article{fds331376,
Author = {Pierce, LB and Yung, PL},
Title = {A polynomial Carleson operator along the
paraboloid},
Journal = {Revista Matemática Iberoamericana},
Publisher = {European Mathematical Society},
Year = {2018},
Key = {fds331376}
}
@article{fds328917,
Author = {Carneiro, E and Madrid, J and Pierce, LB},
Title = {Endpoint Sobolev and BV continuity for maximal
operators},
Journal = {Journal of Functional Analysis},
Volume = {273},
Number = {10},
Pages = {3262-3294},
Publisher = {Elsevier BV},
Year = {2017},
Month = {November},
url = {http://dx.doi.org/10.1016/j.jfa.2017.08.012},
Doi = {10.1016/j.jfa.2017.08.012},
Key = {fds328917}
}
@article{fds328811,
Author = {Pierce, LB and Heath-Brown, DR},
Title = {Averages and moments associated to class numbers of
imaginary quadratic fields},
Volume = {153},
Number = {11},
Pages = {2287-2309},
Publisher = {Oxford University Press (OUP)},
Year = {2017},
Month = {November},
url = {http://dx.doi.org/10.1112/S0010437X1700728X},
Abstract = {For any odd prime l, let hl(-d)denote the l-part of the
class number of the imaginary quadratic field Q(d).
Nontrivial pointwise upper bounds are known only for ;
nontrivial upper bounds for averages of have previously been
known only for averages of hl(-d). In this paper we prove
nontrivial upper bounds for the average of for all
primesl≥7, as well as nontrivial upper bounds for certain
higher moments for all primes ≥3.},
Doi = {10.1112/S0010437X1700728X},
Key = {fds328811}
}
@article{fds330204,
Author = {Pierce, LB},
Title = {The Vinogradov Mean Value Theorem [after Wooley, and
Bourgain, Demeter and Guth]},
Journal = {AstÉrisque},
Publisher = {Centre National de la Recherche Scientifique},
Year = {2017},
Month = {July},
Abstract = {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.},
Key = {fds330204}
}
@article{fds320389,
Author = {Heath-Brown, DR and Pierce, LB},
Title = {Simultaneous integer values of pairs of quadratic
forms},
Journal = {Journal fur die Reine und Angewandte Mathematik},
Volume = {2017},
Number = {727},
Pages = {85-143},
Publisher = {WALTER DE GRUYTER GMBH},
Year = {2017},
Month = {June},
url = {http://dx.doi.org/10.1515/crelle-2014-0112},
Abstract = {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 particular such forms
simultaneously attain prime values if the obvious local
conditions hold. The proof uses the circle method, and in
particular pioneers a two-dimensional version of a
Kloosterman refinement.},
Doi = {10.1515/crelle-2014-0112},
Key = {fds320389}
}
@article{fds320661,
Author = {Guo, S and Pierce, LB and Roos, J and Yung, P},
Title = {Polynomial Carleson operators along monomial curves in the
plane},
Journal = {Journal of Geometric Analysis},
Volume = {27},
Number = {4},
Pages = {2977-3012},
Publisher = {Springer Verlag},
Year = {2017},
url = {http://dx.doi.org/10.1007/s12220-017-9790-7},
Abstract = {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 stopping-time functions that
depend on only one of the two ambient variables. A
motivation for studying these partial operators is the
curious feature that, despite their apparent limitations,
for certain combinations of curve and phase, $L^2$ bounds
for partial operators along curves imply the full strength
of the $L^2$ bound for a one-dimensional Carleson operator,
and for a quadratic Carleson operator. Our methods, which
can at present only treat certain combinations of curves and
phases, in some cases adapt a $TT^*$ method to treat phases
involving fractional monomials, and in other cases use a
known vector-valued variant of the Carleson-Hunt
theorem.},
Doi = {10.1007/s12220-017-9790-7},
Key = {fds320661}
}
@article{fds320660,
Author = {Ellenberg, J and Pierce, LB and Wood, MM},
Title = {On ℓ-torsion in class groups of number
fields},
Journal = {Algebra and Number Theory},
Volume = {11},
Number = {8},
Pages = {1739-1778},
Publisher = {Mathematical Sciences Publishers},
Year = {2017},
url = {http://dx.doi.org/10.2140/ant.2017.11.1739},
Abstract = {© 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 ε {2; 3; 4; 5} (with the
additional restriction in the case d D 4 that the field be
non-D 4 ). For sufficiently large ℓ (specified
explicitly), these results are as strong as a previously
known bound that is conditional on GRH. As part of our
argument, we develop a probabilistic “Chebyshev sieve,”
and give uniform, power-saving error terms for the
asymptotics of quartic (non-D 4 ) and quintic fields with
chosen splitting types at a finite set of
primes.},
Doi = {10.2140/ant.2017.11.1739},
Key = {fds320660}
}
@article{fds320387,
Author = {Pierce, LB},
Title = {Burgess bounds for multi-dimensional short mixed character
sums},
Journal = {Journal of Number Theory},
Volume = {163},
Pages = {172-210},
Publisher = {Elsevier BV},
Year = {2016},
Month = {June},
url = {http://dx.doi.org/10.1016/j.jnt.2015.08.022},
Abstract = {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 characters. We combine a
multi-dimensional Burgess method with recent results on
multi-dimensional Vinogradov Mean Value Theorems for
translation-dilation invariant systems in order to prove
character sum bounds in k≥ 1 dimensions that recapture the
Burgess bound in dimension 1. Moreover, we show that by
embedding any given polynomial f into an advantageously
chosen translation-dilation invariant system constructed in
terms of f, we may in many cases significantly improve the
bound for the associated character sum, due to a novel
phenomenon that occurs only in dimensions k≥
2.},
Doi = {10.1016/j.jnt.2015.08.022},
Key = {fds320387}
}
@article{fds320386,
Author = {Pierce, LB and Schindler, D and Wood, MM},
Title = {Representations of integers by systems of three quadratic
forms},
Journal = {Proceedings of the London Mathematical Society},
Volume = {3},
Number = {113},
Pages = {289-344},
Publisher = {London Mathematical Society},
Year = {2016},
url = {http://dx.doi.org/10.1112/plms/pdw027},
Abstract = {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; reducing the required number of
variables remains a significant open problem. In this work,
we consider the case of 3 forms and improve on the classical
result by reducing the number of required variables to $k
\geq 10$ for "almost all" tuples, under appropriate
nonsingularity assumptions on the forms $Q_1,Q_2,Q_3$. To
accomplish this, we develop a three-dimensional analogue of
Kloosterman's circle method, in particular capitalizing on
geometric properties of appropriate systems of three
quadratic forms.},
Doi = {10.1112/plms/pdw027},
Key = {fds320386}
}
@article{fds302459,
Author = {Bober, J and Carneiro, E and Hughes, K and Kosz, D and Pierce,
LB},
Title = {Corrigendum to “on a discrete version of Tanaka’s
theorem for maximal functions”},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {143},
Number = {12},
Pages = {5471-5473},
Publisher = {American Mathematical Society (AMS)},
Year = {2015},
ISSN = {0002-9939},
url = {http://dx.doi.org/10.1090/proc/12778},
Abstract = {© 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.},
Doi = {10.1090/proc/12778},
Key = {fds302459}
}
@article{fds302460,
Author = {Heath-Brown, DR and Pierce, LB},
Title = {Burgess bounds for short mixed character
sums},
Journal = {Journal of the London Mathematical Society},
Volume = {91},
Number = {3},
Pages = {693-708},
Publisher = {Oxford University Press (OUP)},
Year = {2015},
ISSN = {0024-6107},
url = {http://dx.doi.org/10.1112/jlms/jdv009},
Abstract = {© 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.},
Doi = {10.1112/jlms/jdv009},
Key = {fds302460}
}
@article{fds320388,
Author = {Alaifari, R and Pierce, LB and Steinerberger, S},
Title = {Lower bounds for the truncated Hilbert transform},
Journal = {arXiv:1311.6845 [math]},
Volume = {32},
Number = {1},
Pages = {23-56},
Publisher = {European Mathematical Publishing House},
Year = {2013},
Month = {November},
url = {http://dx.doi.org/10.4171/rmi/880},
Abstract = {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 unique answer (the nullspace is trivial)
but is severely ill-posed. We isolate the difficulty and
show that by restricting $f$ to functions with controlled
total variation, reconstruction becomes stable. In
particular, for functions $f \in H^1(I)$, we show that $$
\|Hf\|_{L^2(J)} \geq c_1 \exp{\left(-c_2
\frac{\|f_x\|_{L^2(I)}}{\|f\|_{L^2(I)}}\right)} \| f
\|_{L^2(I)} ,$$ for some constants $c_1, c_2 > 0$ depending
only on $I, J$. This inequality is sharp, but we conjecture
that $\|f_x\|_{L^2(I)}$ can be replaced by
$\|f_x\|_{L^1(I)}$.},
Doi = {10.4171/rmi/880},
Key = {fds320388}
}
@article{fds302458,
Author = {Pierce, LB},
Title = {Correction to "discrete fractional radon transforms and
quadratic forms," duke math. J. 161 2012,
69-106},
Journal = {Duke Mathematical Journal},
Volume = {162},
Number = {6},
Pages = {1203-1204},
Publisher = {Duke University Press},
Year = {2013},
ISSN = {0012-7094},
url = {http://dx.doi.org/10.1215/00127094-2210146},
Doi = {10.1215/00127094-2210146},
Key = {fds302458}
}
@article{fds302454,
Author = {Bober, J and Carneiro, E and Hughes, K and Pierce,
LB},
Title = {On a discrete version of Tanaka's theorem for maximal
functions},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {140},
Number = {5},
Pages = {1669-1680},
Publisher = {American Mathematical Society (AMS)},
Year = {2012},
Month = {May},
ISSN = {0002-9939},
url = {http://dx.doi.org/10.1090/S0002-9939-2011-11008-6},
Abstract = {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, given a function $f:
\mathbb{Z} \to \mathbb{R}$ of bounded variation,
$$\textrm{Var}(\widetilde{M} f) \leq \textrm{Var}(f),$$
where $\textrm{Var}(f)$ represents the total variation of
$f$. For the discrete centered maximal operator $M$ we prove
that, given a function $f: \mathbb{Z} \to \mathbb{R}$ such
that $f \in \ell^1(\mathbb{Z})$, $$\textrm{Var}(Mf) \leq C
\|f\|_{\ell^1(\mathbb{Z})}.$$ This provides a positive
solution to a question of Haj{\l}asz and Onninen \cite{HO}
in the discrete one-dimensional case.},
Doi = {10.1090/S0002-9939-2011-11008-6},
Key = {fds302454}
}
@article{fds302455,
Author = {Pierce, LB},
Title = {Discrete fractional radon transforms and quadratic
forms},
Journal = {Duke Mathematical Journal},
Volume = {161},
Number = {1},
Pages = {69-106},
Publisher = {Duke University Press},
Year = {2012},
ISSN = {0012-7094},
url = {http://dx.doi.org/10.1215/00127094-1507288},
Abstract = {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 sufficient conditions for them to
extend to bounded operators from l p to l q. The method
involves an intricate spectral decomposition according to
major and minor arcs, motivated by ideas from the circle
method of Hardy and Littlewood. Techniques from harmonic
analysis, in particular Fourier transform methods and
oscillatory integrals, as well as the number theoretic
structure of quadratic forms, exponential sums, and theta
functions, play key roles in the proof.},
Doi = {10.1215/00127094-1507288},
Key = {fds302455}
}
@article{fds302456,
Author = {Heath-Brown, DR and Pierce, LB},
Title = {Counting rational points on smooth cyclic
covers},
Journal = {Journal of Number Theory},
Volume = {132},
Number = {8},
Pages = {1741-1757},
Publisher = {Elsevier BV},
Year = {2012},
ISSN = {0022-314X},
url = {http://dx.doi.org/10.1016/j.jnt.2012.02.010},
Abstract = {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 covers of degree r≥ 3 when n>
10. This is achieved by a new bound for the number of
perfect r-th power values of a polynomial with nonsingular
leading form, obtained via a combination of an r-th power
sieve and the q-analogue of van der Corput's method. © 2012
Elsevier Inc.},
Doi = {10.1016/j.jnt.2012.02.010},
Key = {fds302456}
}
@article{fds302457,
Author = {Pierce, LB},
Title = {A note on discrete fractional integral operators on the
heisenberg group},
Journal = {International Mathematics Research Notices},
Volume = {2012},
Number = {1},
Pages = {17-33},
Publisher = {Oxford University Press (OUP)},
Year = {2012},
ISSN = {1073-7928},
url = {http://dx.doi.org/10.1093/imrn/rnr008},
Abstract = {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 Press. All rights reserved.},
Doi = {10.1093/imrn/rnr008},
Key = {fds302457}
}
@article{fds302453,
Author = {Pierce, LB},
Title = {On discrete fractional integral operators and mean values of
Weyl sums},
Journal = {Bulletin of the London Mathematical Society},
Volume = {43},
Number = {3},
Pages = {597-612},
Publisher = {Oxford University Press (OUP)},
Year = {2011},
ISSN = {0024-6093},
url = {http://dx.doi.org/10.1112/blms/bdq127},
Abstract = {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 interactions between the
Fourier multiplier and mean values of Weyl sums. These mean
values express the average behaviour of the number rs, k(l)
of representations of a positive integer l as a sum of s
positive kth powers. Recent deep results within the context
of Waring's problem and Weyl sums enable us to prove a
further range of complementary results for the discrete
operator under consideration. © 2011 London Mathematical
Society.},
Doi = {10.1112/blms/bdq127},
Key = {fds302453}
}
@article{fds302451,
Author = {Pierce, LB},
Title = {A note on twisted discrete singular Radon
transforms},
Journal = {Mathematical Research Letters},
Volume = {17},
Number = {4},
Pages = {701-720},
Publisher = {International Press of Boston},
Year = {2010},
ISSN = {1073-2780},
url = {http://dx.doi.org/10.4310/mrl.2010.v17.n4.a10},
Abstract = {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 oscillatory component, via a simple method of
descent argument. Second, we note an ℓ2 bound for
quasi-translation invariant discrete twisted Radon
transforms. Finally, we extend an existing ℓ2 bound for a
closely related non-translation invariant discrete
oscillatory integral operator with singular kernel to an
ℓp bound for all 1 < p < 1∞. This requires an
intricate induction argument involving layers of
decompositions of the operator according to the Diophantine
properties of the coefficients of its polynomial phase
function. Copyright © 2010 International
Press.},
Doi = {10.4310/mrl.2010.v17.n4.a10},
Key = {fds302451}
}
@article{fds302452,
Author = {Pierce, LB},
Title = {A bound for the 3-part of class numbers of quadratic fields
by means of the square sieve},
Journal = {Forum Mathematicum},
Volume = {18},
Number = {4},
Pages = {677-698},
Publisher = {WALTER DE GRUYTER GMBH},
Year = {2006},
ISSN = {0933-7741},
url = {http://dx.doi.org/10.1515/FORUM.2006.034},
Abstract = {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-free positive integer d and bounded
x, z. © de Gruyter 2006.},
Doi = {10.1515/FORUM.2006.034},
Key = {fds302452}
}
@article{fds302450,
Author = {Pierce, LB},
Title = {The 3-part of class numbers of quadratic
fields},
Journal = {Journal of the London Mathematical Society},
Volume = {71},
Number = {3},
Pages = {579-598},
Publisher = {Oxford University Press (OUP)},
Year = {2005},
ISSN = {0024-6107},
url = {http://dx.doi.org/10.1112/S002461070500637X},
Abstract = {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 congruence xa,≡, yb modulo q in
the ranges x ≤ X and y ≤ Y, where a,b are nonzero
integers and q is a square-free positive integer.
Furthermore, it is shown that the number of elliptic curves
over ℚ with conductor N is O(N55/112+ε)in general and
O(N5/12+ε) if N has a divisor of size N5/6. These results
are the first improvements to the trivial bound O(|D|
1/2+ε) and the resulting bound O(N1/2+ε) for the 3-part
and the number of elliptic curves, respectively. © 2005
London Mathematical Society.},
Doi = {10.1112/S002461070500637X},
Key = {fds302450}
}
%% Other
@misc{fds299989,
Author = {L.B. Pierce},
Title = {Recent Publications},
Year = {2015},
Key = {fds299989}
}
|