Combinatorial and additive number theory III CANT PDF
Preview Combinatorial and additive number theory III CANT
Springer 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 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 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 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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].
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