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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
             imaginary quadratic fields},
   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
             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}
}

 

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