Table Of ContentSpringer Proceedings in Mathematics & Statistics
Melvyn B. Nathanson Editor
Combinatorial
and Additive
Number Theory
III
CANT, New York, USA, 2017 and 2018
Springer Proceedings in Mathematics &
Statistics
Volume 297
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Melvyn B. Nathanson
Editor
Combinatorial and Additive
Number Theory III
CANT, New York, USA, 2017 and 2018
123
Editor
Melvyn B. Nathanson
Department ofMathematics
Lehman Collegeandthe Graduate Center
City University of NewYork
NewYork,NY, USA
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-3-030-31105-6 ISBN978-3-030-31106-3 (eBook)
https://doi.org/10.1007/978-3-030-31106-3
Mathematics Subject Classification (2010): 03H15, 11B05, 11B13, 11B75, 11D07, 11D25, 11E04,
14H05,15A12,15B51
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Preface
Workshops on Combinatorial and Additive Number Theory (CANT) have been
organized atthe CUNYGraduate Center inNew York every year since 2003. The
4-day CANT conferences are held in May, usually from Tuesday to Friday of the
week immediately preceding or immediately following Memorial Day. They have
become a fixed point in the number theory calendar.
These workshops are arranged by the New York Number Theory Seminar. The
seminarwasstartedin1981byDavidandGregoryChudnovsky,HarveyCohn,and
Melvyn B. Nathanson, and for 38 years has been meeting at the CUNY Graduate
CentereveryThursdayafternoonduringtheacademicyear,andalsointhesummer.
This volume contains papers presented at the CANT 2017 and CANT 2018
workshops. There are 17 papers on important topics in number theory and related
parts of mathematics. These topics include sumsets, partitions, convex polytopes
and discrete geometry, Ramsey theory, commutative algebra and arithmetic
geometry, and applications of logic and nonstandard analysis to number theory.
I thank the Number Theory Foundation, Springer, and the Journal of Number
Theory (Elsevier) for their support of CANT.
I am grateful to Springer and to mathematics editor Dahlia Fisch for making
possible the publication of the proceedings of the CANT 2017 and CANT 2018
workshops.
Previous volumes are [1] and [2].
New York, USA Melvyn B. Nathanson
References
1. M.B.Nathanson,editor,CombinatorialandAdditiveNumberTheory–CANT2011and2012,
SpringerProc.Math.Stat.,vol.101,Springer,NewYork,2014.
2. M. B. Nathanson, editor, Combinatorial and Additive Number Theory II–CANT 2015 and
2016,SpringerProc.Math.Stat.,vol.220,Springer,NewYork,2017.
v
Contents
Weighted Zero-Sums for Some Finite Abelian Groups
of Higher Ranks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
S. D. Adhikari, Bidisha Roy and Subha Sarkar
Counting Monogenic Cubic Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Shabnam Akhtari
The Zeckendorf Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Paul Baird-Smith, Alyssa Epstein, Kristen Flint and Steven J. Miller
Iterated Riesel and Iterated Sierpiński Numbers . . . . . . . . . . . . . . . . . . 39
Holly Paige Chaos and Carrie E. Finch-Smith
A General Framework for Studying Finite Rainbow
Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Mike Desgrottes, Steven Senger, David Soukup and Renjun Zhu
Translation Invariant Filters and van der Waerden’s Theorem. . . . . . . 65
Mauro Di Nasso
Central Values for Clebsch–Gordan Coefficients . . . . . . . . . . . . . . . . . . 75
Robert W. Donley Jr.
Numerical Semigroups Generated by Squares and Cubes
of Three Consecutive Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Leonid G. Fel
On Supra-SIM Sets of Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . 123
Isaac Goldbring and Steven Leth
Mean Row Values in (u, v)-Calkin–Wilf Trees. . . . . . . . . . . . . . . . . . . . 133
Sandie Han, Ariane M. Masuda, Satyanand Singh and Johann Thiel
Dimensions of Monomial Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Melvyn B. Nathanson
vii
viii Contents
Matrix Scaling Limits in Finitely Many Iterations . . . . . . . . . . . . . . . . . 161
Melvyn B. Nathanson
Not All Groups Are LEF Groups, or Can You Know If a Group
Is Infinite? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Melvyn B. Nathanson
Binary Quadratic Forms in Difference Sets . . . . . . . . . . . . . . . . . . . . . . 175
Alex Rice
Egyptian Fractions, Nonstandard Extensions of RR, and Some
Diophantine Equations Without Many Solutions . . . . . . . . . . . . . . . . . . 197
David A. Ross
A Dual-Radix Approach to Steiner’s 1-Cycle Theorem . . . . . . . . . . . . . 209
Andrey Rukhin
Potentially Stably Rational Del Pezzo Surfaces over
Nonclosed Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Yuri Tschinkel and Kaiqi Yang
Weighted Zero-Sums for Some Finite
Abelian Groups of Higher Ranks
S.D.Adhikari,BidishaRoyandSubhaSarkar
Abstract Inthisarticle,weconsiderthestudyoftheDavenportconstantwithweight
{±1}forsomefiniteabeliangroupsofhigherranksandtakeupsomerelatedques-
tions.Forinstance,weshowthatforanoddprimep,anysequenceoverG =(Z/pZ)3
oflength4p−3whichcontainsatleastfivezero-elements,thereisa{±1}-weighted
zero-sum subsequence of length p. We also show that for an odd prime p and for
apositiveevenintegerk ≥2whichdivides p−1,ifθ isanelementoforderk of
themultiplicativegroup(Z/pZ)∗and Aisthesubgroupof(Z/pZ)∗generatedbyθ,
thenanysequenceover(Z/pZ)k+1oflength4p+ p−1 −1containsan A-weighted
k
zero-sumsubsequenceoflength3p.Intheintroduction,wegiveasmallexpository
accountoftheareaandmentionsomerelevantexpositoryarticles.
1 Introduction
LetG beafiniteabeliangroup(writtenadditively)andletexp(G)betheexponent
ofthegroupG.ByasequenceoverG wemeanafinitesequenceofelementsfrom
G inwhichrepetitionoftermsisallowed.Inthiswaywecanviewasequenceasan
elementofthefreeabelianmonoidF(G)withmultiplicativenotation.
We call a sequence S =g g ...g ∈F(G) to be a zero-sum sequence if g +
1 2 k 1
g +···+g =0where0istheidentityelementofG.
2 k
ForanabeliangroupG,theDavenportconstant D(G)isdefinedtobetheleast
positive integer (cid:3) such that if we take any sequence of length (cid:3) from G, there is a
B
S.D.Adhikari( )
(FormerlyatHarish-ChandraResearchInstitute)DepartmentofMathematics,Ramakrishna
MissionVivekanandaEducationalandResearchInstitute,Belur711202,India
e-mail:[email protected]
B.Roy·S.Sarkar
Harish-ChandraResearchInstitute,HBNI,Jhunsi,Allahabad,India
e-mail:[email protected]
S.Sarkar
e-mail:[email protected]
©SpringerNatureSwitzerlandAG2020 1
M.B.Nathanson(ed.),CombinatorialandAdditiveNumberTheoryIII,
SpringerProceedingsinMathematics&Statistics297,
https://doi.org/10.1007/978-3-030-31106-3_1
2 S.D.Adhikarietal.
non-emptyzero-sumsubsequence.Theearlymotivationforthestudyofthisconstant
[33]wasfactorizationinalgebraicnumberfields.Laterthisconstantfoundimportant
rolesingraphtheory(seeforinstance,[13]or[19])andintheproofoftheinfinitude
ofCarmichaelnumbersbyAlfordetal.[10].
Given a finite abelian group G =(cid:2)(Z/n1Z)×(Z/n2Z)×···×(Z/ndZ) with
n |n |···|n , writing M(G)=1+ d (n −1), it is trivial to see that M(G)≤
1 2 d i=1 i
D(G)≤|G|.Theequality D(G)=|G|holdsifandonlyifG =Z/nZ,thecyclic
group of order n. Olson [30, 31] proved that D(G)= M(G) for all finite abelian
groups of rank 2 and for all p-groups. It is also known that D(G)> M(G) for
infinitelymanyfiniteabeliangroupsofrankd >3(see[21],forinstance).
ThebestknownboundisduetovanEmdeBoasandKruyswijk[12]whoproved
that (cid:3) (cid:4)
|G|
D(G)≤n 1+log , (1)
n
wherenistheexponentofG.ThiswasagainprovedbyAlfordetal.[10].
Westatethefollowingconjectures:
1. We have D(G)= M(G) for all G with rank d =3 or G =(Z/nZ)d [20] and
[18].
2. F(cid:2)or G =(Z/n1Z)×(Z/n2Z)×···×(Z/ndZ) with n1|n2|···|nd, D(G)≤
d n [28].
i=1 i
ForanabeliangroupGwithexp(G)=n,theErdo˝s–Ginzburg–Zivconstants(G)
isdefinedtobetheleastpositiveinteger(cid:3)suchthatifwetakeanysequenceoflength
(cid:3)overG thereisazero-sumsubsequenceoflengthn.
ThenameErdo˝s–Ginzburg–Zivconstantisaftertheprototypeofzero-sumresult
[17]byErdo˝s,GinzburgandZiv,whereitwasprovedthats(Z/nZ)≤2n−1.The
exampleofthesequence(0,0,...,0,1,1,...,1)oflength(2n−2)havingnozero-
(cid:5) (cid:6)(cid:7) (cid:8) (cid:5) (cid:6)(cid:7) (cid:8)
n−1 n−1
sumsubsequenceoflengthn,establishesthats(Z/nZ)=2n−1.
ForthegroupG =(Z/nZ)2,Kemnitz[27]hadconjecturedthats(G)=4n−3.
In2000,Rónyai[34]cameveryclosetoitbyprovingthats((Z/pZ)2)≤4p−2,
foraprime pandfinallytheconjecturewasconfirmedbyReiher[32]in2007.
Tillnowtheexactvalueoftheconstants(G)whereG =(Z/nZ)d andd ≥3is
unknown.Foralloddintegersn,Elsholtz[16]provedalowerboundas
s((Z/nZ)d)≥(1·125)(cid:6)d3(cid:7)(n−1)2d +1.
In the other direction, Alon and Dubiner [11] proved that there is an absolute
constantc>0sothatforalln,
s((Z/nZ)d)≤(cdlog d)dn.
2
Forfurtherreadingsinthisdirectionwerefertothefollowingarticles[8, 11, 13,
15, 19].
