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## Publications of Lillian B. Pierce    :chronological  alphabetical  combined listing:

%% Papers Published
@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 = {Heath-Brown, DR and Pierce, LB},
Title = {Averages and moments associated to class numbers of
Journal = {Compositio Mathematica},
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 = {© 2017 The Authors. 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{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 = {(Submitted)},
Year = {2017},
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.},
Key = {fds330203}
}

@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 & 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 = {© 2015 Elsevier Inc. 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
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
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},
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&gt;
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
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
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 &lt; p &lt; 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
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}
}



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