Combinatorial and additive number theory III, 1st ed , melvyn b nathanson, 2020 2893

237 0 0
  • Loading ...
1/237 trang
Tải xuống

Thông tin tài liệu

Ngày đăng: 08/05/2020, 06:55

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 of Mathematics Lehman College and the Graduate Center City University of New York New York, NY, USA ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-030-31105-6 ISBN 978-3-030-31106-3 (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 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part 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 or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Workshops on Combinatorial and Additive Number Theory (CANT) have been organized at the CUNY Graduate Center in New 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 seminar was started in 1981 by David and Gregory Chudnovsky, Harvey Cohn, and Melvyn B Nathanson, and for 38 years has been meeting at the CUNY Graduate Center every Thursday afternoon during the academic year, and also in the summer 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 M B Nathanson, editor, Combinatorial and Additive Number Theory–CANT 2011 and 2012, Springer Proc Math Stat., vol 101, Springer, New York, 2014 M B Nathanson, editor, Combinatorial and Additive Number Theory II–CANT 2015 and 2016, Springer Proc Math Stat., vol 220, Springer, New York, 2017 v Contents Weighted Zero-Sums for Some Finite Abelian Groups of Higher Ranks S D Adhikari, Bidisha Roy and Subha Sarkar Counting Monogenic Cubic Orders Shabnam Akhtari 13 The Zeckendorf Game Paul Baird-Smith, Alyssa Epstein, Kristen Flint and Steven J Miller 25 Iterated Riesel and Iterated Sierpiński Numbers Holly Paige Chaos and Carrie E Finch-Smith 39 A General Framework for Studying Finite Rainbow Configurations Mike Desgrottes, Steven Senger, David Soukup and Renjun Zhu 55 Translation Invariant Filters and van der Waerden’s Theorem Mauro Di Nasso 65 Central Values for Clebsch–Gordan Coefficients Robert W Donley Jr 75 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 R, 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, Bidisha Roy and Subha Sarkar Abstract In this article, we consider the study of the Davenport constant with weight {±1} for some finite abelian groups of higher ranks and take up some related questions For instance, we show that for an odd prime p, any sequence over G = (Z/ pZ)3 of length p − which contains at least five zero-elements, there is a {±1}-weighted zero-sum subsequence of length p We also show that for an odd prime p and for a positive even integer k ≥ which divides p − 1, if θ is an element of order k of the multiplicative group (Z/ pZ)∗ and A is the subgroup of (Z/ pZ)∗ generated by θ , − contains an A-weighted then any sequence over (Z/ pZ)k+1 of length p + p−1 k zero-sum subsequence of length p In the introduction, we give a small expository account of the area and mention some relevant expository articles Introduction Let G be a finite abelian group (written additively) and let exp(G) be the exponent of the group G By a sequence over G we mean a finite sequence of elements from G in which repetition of terms is allowed In this way we can view a sequence as an element of the free abelian monoid F(G) with multiplicative notation We call a sequence S = g1 g2 gk ∈ F(G) to be a zero-sum sequence if g1 + g2 + · · · + gk = where is the identity element of G For an abelian group G, the Davenport constant D(G) is defined to be the least positive integer such that if we take any sequence of length from G, there is a S D Adhikari (B) (Formerly at Harish-Chandra Research Institute) Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur 711202, India e-mail: adhikari@hri.res.in B Roy · S Sarkar Harish-Chandra Research Institute, HBNI, Jhunsi, Allahabad, India e-mail: bidisharoy@hri.res.in S Sarkar e-mail: subhasarkar@hri.res.in © Springer Nature Switzerland AG 2020 M B Nathanson (ed.), Combinatorial and Additive Number Theory III, Springer Proceedings in Mathematics & Statistics 297, https://doi.org/10.1007/978-3-030-31106-3_1 S D Adhikari et al non-empty zero-sum subsequence The early motivation for the study of this constant [33] was factorization in algebraic number fields Later this constant found important roles in graph theory (see for instance, [13] or [19]) and in the proof of the infinitude of Carmichael numbers by Alford et al [10] Given a finite abelian group G = (Z/n Z) × (Z/n Z) × · · · × (Z/n d Z) with d (n i − 1), it is trivial to see that M(G) ≤ n |n | · · · |n d , writing M(G) = + i=1 D(G) ≤ |G| The equality D(G) = |G| holds if and only if G = Z/nZ, the cyclic group of order n Olson [30, 31] proved that D(G) = M(G) for all finite abelian groups of rank and for all p-groups It is also known that D(G) > M(G) for infinitely many finite abelian groups of rank d > (see [21], for instance) The best known bound is due to van Emde Boas and Kruyswijk [12] who proved that |G| , (1) D(G) ≤ n + log n where n is the exponent of G This was again proved by Alford et al [10] We state the following conjectures: We have D(G) = M(G) for all G with rank d = or G = (Z/nZ)d [20] and [18] For G = (Z/n Z) × (Z/n Z) × · · · × (Z/n d Z) with n |n | · · · |n d , D(G) ≤ d i=1 n i [28] For an abelian group G with exp(G) = n, the Erd˝os–Ginzburg–Ziv constant s(G) is defined to be the least positive integer such that if we take any sequence of length over G there is a zero-sum subsequence of length n The name Erd˝os–Ginzburg–Ziv constant is after the prototype of zero-sum result [17] by Erd˝os, Ginzburg and Ziv, where it was proved that s(Z/nZ) ≤ 2n − The example of the sequence (0, 0, , 0, 1, 1, , 1) of length (2n − 2) having no zeron−1 n−1 sum subsequence of length n, establishes that s(Z/nZ) = 2n − For the group G = (Z/nZ)2 , Kemnitz [27] had conjectured that s(G) = 4n − In 2000, Rónyai [34] came very close to it by proving that s((Z/ pZ)2 ) ≤ p − 2, for a prime p and finally the conjecture was confirmed by Reiher [32] in 2007 Till now the exact value of the constant s(G) where G = (Z/nZ)d and d ≥ is unknown For all odd integers n, Elsholtz [16] proved a lower bound as s((Z/nZ)d ) ≥ (1 · 125) d (n − 1)2d + In the other direction, Alon and Dubiner [11] proved that there is an absolute constant c > so that for all n, s((Z/nZ)d ) ≤ (cd log2 d)d n For further readings in this direction we refer to the following articles [8, 11, 13, 15, 19] A Dual-Radix Approach to Steiner’s 1-Cycle Theorem τ 219 (−)w E w Tw = 0≤w
- Xem thêm -

Xem thêm: Combinatorial and additive number theory III, 1st ed , melvyn b nathanson, 2020 2893 , Combinatorial and additive number theory III, 1st ed , melvyn b nathanson, 2020 2893

Mục lục

Xem thêm

Gợi ý tài liệu liên quan cho bạn