Combinatorial and Additive Number Theory: CANT 2011 and 2012

309 Pages·2014·2.93 MB·English

by  Melvyn B. Nathanson (eds.)

Springer Proceedings in Mathematics & Statistics
Melvyn B. Nathanson Editor
Combinatorial
and Additive
Number Theory
CANT 2011 and 2012
Springer Proceedings in Mathematics & Statistics
Volume 101
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Melvyn B. Nathanson
Editor
Combinatorial and Additive
Number Theory
CANT 2011 and 2012
123
Editor
MelvynB.Nathanson
DepartmentofMathematics
LehmanCollege(CUNY)
Bronx,NY,USA
ISSN2194-1009 ISSN2194-1017(electronic)
ISBN978-1-4939-1600-9 ISBN978-1-4939-1601-6(eBook)
DOI10.1007/978-1-4939-1601-6
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2014950047
MathematicsSubjectClassification(2010):05-06,11-06,05A17,05A18,05A19,05C35,11B05,11B13,
11B30,11B75,20M05,20F69,20F65
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Preface
ThisvolumecontainsselectedpapersfromtworecentconferencesonCombinatorial
andAdditiveNumberTheory(CANT2011andCANT2012).
These meetings, which started in 2003, take place every year in May at the
CUNYGraduateCenterinNewYork.Ithasbecometraditionaltoincludenotonly
alargenumberofcontributedtalksbutalsoatleastoneseriesoftalksbyaninvited
lecturer.Speakersareencouragednottopreparehighlytechnicaltalksintendedonly
forthe“experts,”butto includeexpositoryandmotivationalmaterialandto tryto
maketheirlecturescomprehensibletograduatestudentswhoarejustbeginningto
think about research. For this reason, the conferences also include daily problem
anddiscussionsessions.
It is a pleasureto acknowledgethe supportof the NumberTheoryFoundation,
which has, for many years, facilitated the participation of undergraduate and
graduatestudentsintheseconferences.
MelvynB.Nathanson
v

Contents
GeneralizedRamanujanPrimes................................................ 1
NadineAmersi,OliviaBeckwith,StevenJ.Miller,RyanRonan,
andJonathanSondow
ArithmeticCongruenceMonoids:ASurvey.................................. 15
PaulBaginskiandScottChapman
AShortProofofKneser’sAdditionTheoremforAbelianGroups......... 39
MattDeVos
LowerandUpperClassesofNaturalNumbers............................... 43
L.HaddadandC.Helou
TheProbabilityThatRandomPositiveIntegersAre3-Wise
RelativelyPrime.................................................................. 55
JerryHu
Sharpness ofFalconer’sEstimate andthe SingleDistance
ProbleminZd .................................................................... 63
q
AlexIosevichandStevenSenger
FindingandCountingMSTDSets............................................. 79
GeoffreyIyer,OlegLazarev,StevenJ.Miller,andLiyangZhang
DensityVersionsofPlünneckeInequality:Epsilon-DeltaApproach....... 99
RenlingJin
ProblemsandResultsonIntersectiveSets .................................... 115
TháiHoàngLê
PolynomialDifferencesinthePrimes.......................................... 129
NeilLyallandAlexRice
vii
viii Contents
MostSubsetsAreBalancedinFiniteGroups................................. 147
StevenJ.MillerandKevinVissuet
GaussianBehaviorinGeneralizedZeckendorfDecompositions............ 159
StevenJ.MillerandYinghuiWang
Additive Number Theory and Linear Semigroups
withIntermediateGrowth ...................................................... 175
MelvynB.Nathanson
AdjoiningIdentitiesandZerostoSemigroups................................ 195
MelvynB.Nathanson
On the Grothendieck Group Associated to Solutions
of a Functional Equation Arising from Multiplication
ofQuantumIntegers............................................................. 203
LanNguyen
ThePlünnecke–RuzsaInequality:AnOverview ............................. 229
G.Petridis
LerchQuotients,LerchPrimes,Fermat-WilsonQuotients,
andtheWieferich-Non-WilsonPrimes2,3,14771........................... 243
JonathanSondow
OnSumsRelatedtoCentralBinomialandTrinomialCoefficients ........ 257
Zhi-WeiSun
Generalized Ramanujan Primes
NadineAmersi,OliviaBeckwith,StevenJ.Miller,RyanRonan,
andJonathanSondow
Abstract In 1845, Bertrand conjectured that for all integers x (cid:2) 2, there exists
at least one prime in .x=2;x(cid:2). This was proved by Chebyshev in 1860 and then
generalized by Ramanujan in 1919. He showed that for any n (cid:2) 1, there is a
(smallest) prime R such that (cid:3).x/ (cid:3) (cid:3).x=2/ (cid:2) n for all x (cid:2) R . In 2009
n n
Sondow called R the nth Ramanujan prime and proved the asymptotic behavior
n
R (cid:4) p (where p is the mth prime). He and Laishram proved the bounds
n 2n m
p < R < p , respectively, for n > 1. In the present paper, we generalize
2n n 3n
the intervalof interest by introducinga parameterc 2 .0;1/ and defining the nth
c-Ramanujan prime as the smallest integer R such that for all x (cid:2) R , there
c;n c;n
are at least n primes in .cx;x(cid:2). Using consequences of strengthened versions of
the Prime Number Theorem, we prove that R exists for all n and all c, that
c;n
Rc;n (cid:4) p n as n ! 1, and that the fraction of primeswhich are c-Ramanujan
1(cid:2)c
convergesto1(cid:3)c.Wethenstudyfinerquestionsrelatedtotheirdistributionamong
theprimesandseethatthec-Ramanujanprimesdisplaystrikingbehavior,deviating
significantlyfromaprobabilisticmodelbasedonbiasedcoinflipping.Thismodel
isrelatedtotheCramermodel,whichcorrectlypredictsmanypropertiesofprimes
onlargescalesbuthasbeenshowntofailinsomeinstancesonsmallerscales.
N.Amersi
DepartmentofMathematics,UniversityCollegeLondon,GowerStreet,
LondonWC1E6BT,UK
e-mail:[e-mail]
O.Beckwith
DepartmentofMathematics,EmoryUniversity,404DowmanDrive,Atlanta,GA30322
e-mail:[e-mail]
S.J.Miller((cid:2))
DepartmentofMathematicsandStatistics,WilliamsCollege,Williamstown,MA01267,USA
e-mail:[e-mail];[e-mail]
R.Ronan
DepartmentofElectricalEngineering,CooperUnion,NewYork,NY10003,USA
e-mail:[e-mail]
J.Sondow
209West97thStreet,NewYork,NY10025,USA
e-mail:[e-mail]
©SpringerScience+BusinessMediaNewYork2014 1
M.B.Nathanson(ed.),CombinatorialandAdditiveNumberTheory,Springer
ProceedingsinMathematics&Statistics101,DOI10.1007/978-1-4939-1601-6__1