Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016 PDF
Preview Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016
Springer Proceedings in Mathematics & Statistics Melvyn B. Nathanson Editor Combinatorial and Additive Number Theory II CANT, New York, NY, USA, 2015 and 2016 Springer Proceedings in Mathematics & Statistics Volume 220 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 II CANT, New York, NY, USA, 2015 and 2016 123 Editor Melvyn B. Nathanson Department ofMathematics Lehman College(CUNY) Bronx, NY USA ISSN 2194-1009 ISSN 2194-1017 (electronic) SpringerProceedings in Mathematics& Statistics ISBN978-3-319-68030-9 ISBN978-3-319-68032-3 (eBook) https://doi.org/10.1007/978-3-319-68032-3 LibraryofCongressControlNumber:2017956728 Mathematics Subject Classification (2010): 05A17, 11A05, 11A41, 11A51, 11B05, 11B13, 11B75, 11N05,11P21,11P81,11P84,11Y05 ©SpringerInternationalPublishingAG2017 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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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The CUNY Graduate Center Workshops on Combinatorial and Additive Number Theory (CANT) have been organized every year, beginning in 2003, by the New York Number Theory Seminar. The seminar was started in 1981 by David and GregoryChudnovsky,HarveyCohn,andMelNathanson,andfor36yearshasbeen meeting every Thursday afternoon during the academic year, and also in the summer. Thefour-dayCANTconferencesareheldinMayattheCUNYGraduateCenter in Manhattan, usually from Tuesday to Friday of the week immediately preceding Memorial Day. They have become a fixed point in the number theory calendar. This collection derives from talks at the CANT 2015 and CANT 2016 work- shops. There are 20 papers on important topics in number theory and related parts of mathematics. These topics include sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, primality testing, and cryptography. IamgratefultoSpringeranditsmathematicseditor,MarcStrauss,forpublishing the proceedings of these meetings. A previous volume is [1]. Bronx, NY, USA Melvyn B. Nathanson Reference 1. M.B. Nathanson, editor, Combinatorial and additive number theory–CANT 2011 and 2012. Springer Proc. Math. Stat. vol.101, Springer, New York, 2014 v Contents OnaConjectureofFoxandKleitmanontheDegreeofRegularityofa Certain Linear Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sukumar Das Adhikari and Shalom Eliahou Open Problems About Sumsets in Finite Abelian Groups: Minimum Sizes and Critical Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Béla Bajnok Benford Behavior of Generalized Zeckendorf Decompositions. . . . . . . . 25 Andrew Best, Patrick Dynes, Xixi Edelsbrunner, Brian McDonald, Steven J. Miller, Kimsy Tor, Caroline Turnage-Butterbaugh and Madeleine Weinstein Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Andrew Best, Karen Huan, Nathan McNew, Steven J. Miller, Jasmine Powell, Kimsy Tor and Madeleine Weinstein Recurrence Identities of b-ary Partitions . . . . . . . . . . . . . . . . . . . . . . . . 53 Dakota Blair CryptographicHashFunctionsandSomeApplicationstoInformation Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Lisa Bromberg Numerical Sets, Core Partitions, and Integer Points in Polytopes . . . . . 99 Hannah Constantin, Ben Houston-Edwards and Nathan Kaplan Pairs of Dot Products in Finite Fields and Rings . . . . . . . . . . . . . . . . . . 129 David Covert and Steven Senger Characteristic, Counting, and Representation Functions Characterized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Charles Helou vii viii Contents Partitions into Parts Simultaneously Regular, Distinct, And/or Flat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 William J. Keith White’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Mizan R. Khan and Karen M. Rogers A Misère-Play H-Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Matthieu Dufour, Silvia Heubach and Urban Larsson A New Proof of Khovanskiĭ’s Theorem on the Geometry of Sumsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Jaewoo Lee Initial Sums of the Legendre Symbol: Is min + max ≥ 0 ?. . . . . . . . . . 205 Kieren MacMillan and Jonathan Sondow A Second Wave of Expanders in Finite Fields . . . . . . . . . . . . . . . . . . . . 215 Brendan Murphy and Giorgis Petridis Sumsets Contained in Sets of Upper Banach Density 1 . . . . . . . . . . . . . 239 Melvyn B. Nathanson The Erdős Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Melvyn B. Nathanson Limits and Decomposition of de Bruijn’s Additive Systems . . . . . . . . . . 255 Melvyn B. Nathanson Extending Babbage’s (Non-)Primality Tests. . . . . . . . . . . . . . . . . . . . . . 269 Jonathan Sondow Conjectures on Representations Involving Primes . . . . . . . . . . . . . . . . . 279 Zhi-Wei Sun On a Conjecture of Fox and Kleitman on the Degree of Regularity of a Certain Linear Equation SukumarDasAdhikariandShalomEliahou Abstract FoxandKleitmanprovedin2006thatforanypositiveintegerb,the2n- variable equation x1+···+xn −xn+1−···−x2n = b is not 2n-regular. More- over,theyconjecturedtheexistenceofanintegerb ≥1suchthatforb=b ,this n n equation is (2n−1)-regular. In this note, we settle the first nontrivial case of the conjecture,namelyforn =2,andweproposeaslightrefinementofit. · · Keywords Partitionregularity Diophantineequation Finitecoloring Monochromaticsolution 1 Introduction Here, Z denotes the set of integers and N+ the set of positive integers. For given integersα ,...,α andc,considerthelinearDiophantineequation L: 1 k (cid:2)k α x = c. i i i=1 FollowingRado[5],givenn ∈N+,equation L issaidtoben-regular if,forevery n-coloringofN+,thereexistsamonochromaticsolutionx ∈Nk+to L. S.DasAdhikari Harish-ChandraResearchInstitute,ChhathnagRoad,Jhunsi,Allahabad211019,India e-mail:[email protected] B S.Eliahou ( ) EA2597-LMPA-LaboratoiredeMathématiquesPuresetAppliquéesJosephLiouville, UniversitéduLittoralCôted’Opale(ULCO),CS,62228Calais,France e-mail:[email protected] S.Eliahou CNRSFR2956,Paris,France ©SpringerInternationalPublishingAG2017 1 M.B.Nathanson(ed.),CombinatorialandAdditiveNumberTheoryII, SpringerProceedingsinMathematics&Statistics220, https://doi.org/10.1007/978-3-319-68032-3_1 2 S.DasAdhikariandS.Eliahou The degree of regularity of L is the largest integer n ≥0, if any, such that L is n-regular. This (possibly infinite) number is denoted by dor(L). If dor(L)=∞, then L issaidtoberegular. A conjecture of Rado [5] states that there is a functionr: N+ →N+ such that givenanyn ∈N+andanyequationα1x1+···+αnxn =0withintegercoefficients, ifthisequationisnotregularoverN+,thenitalreadyfailstober(n)-regular.Even though there is a more general version, we state it here for a single homogeneous equation, as it has been proved by Rado that if the conjecture is true for a single equation, then it is true for a system of finitely many linear equations [5], and as FoxandKleitmanhaveshownthatiftheconjectureistrueforalinearhomogeneous equation,thenitistrueforanylinearequation[3].ThisconjectureisknownasRado’s BoundednessConjecture.Thefirstnontrivialcaseoftheconjecturehasbeenproved byFoxandKleitman[3];moreprecisely,theyestablishedtheboundr(3)≤24.In thesamepaper,theauthorsmadethefollowingconjectureforaveryspecificlinear Diophantineequation[3]. Conjecture1 Let n ≥1. There exists an integer b ≥1 such that the degree of n regularityofthe2n-variableequation x1+···+xn −xn+1−···−x2n = bn isexactly2n−1. Iftrue,thatwouldbebestpossible,sincetheyprovedinthesamepaperthatfor anybn ∈N+,theaboveequationisnot2n-regular. Inthisnote,wesettlethefirstnontrivialcaseoftheconjecture,namelythecasen = 2.Indeed,weshallshowthatifb isanypositivemultipleof6,thenthecorresponding 2 equationhasdegreeofregularityexactly3. Moregenerally,weshalldeterminethedegreeofregularityoverN+oftheequation x +x −x −x = b 1 2 3 4 forallb∈N+.SeeTheorem1fortheexactstatement. A related conjecture of Rado [5],stating that for every positive integer n, there existsalinearhomogeneousequationwithdegreeofregularityequalton,wasproved byAlexeevandTsimerman[1].Beforethatpaper,FoxandRadoic´ic˘ [2]hadshown thatforn ≥2,theequation x1+2x2+···+2n−2xn−1−2n−1xn =0 (1) isnotn-regularandhadconjecturedthatitis(n−1)-regular;Golowich[4]proved theirconjecture,thusprovidinganotherproofoftheabove-mentionedconjectureof Rado.
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