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Zero-sum problem

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578: 526: 114: 158: 505: 619: 427: 418:; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse). Basel: Birkhäuser. pp.  414:. Advanced Courses in Mathematics CRM Barcelona. Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; 462: 406:
Geroldinger, Alfred (2009). "Additive group theory and non-unique factorizations". In Geroldinger, Alfred; Ruzsa, Imre Z. (eds.).
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Reiher, Christian (2007), "On Kemnitz' conjecture concerning lattice-points in the plane",
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Erdős, Paul; Ginzburg, A.; Ziv, A. (1961). "Theorem in the additive number theory".
510: 78: 515:"Covering Systems, Restricted Sumsets, Zero-sum Problems and their Unification" 377: 327: 358: 163: 514: 182:− 2. (Indeed, the lower bound is easy to see: the multiset containing 40: 447:
Additive Number Theory: Inverse Problems and the Geometry of Sumsets
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after its discoverers. It may also be deduced from the
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The classic result in this area is the 1961 theorem of
545:" N.W. Sauer (ed.) R.E. Woodrow (ed.) B. Sands (ed.), 209:
More general results than this theorem exist, such as
178:, but that the same is not true of multisets of size 2 543:
Zero-sum trees: a survey of results and open problems
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Combinatorial number theory and additive group theory
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Finite and Infinite Combinatorics in Sets and Logic
533:Zero-Sum Ramsey Theory: Graphs, Sequences and More 152: 108: 613: 8: 35:. Concretely, given a finite abelian group 174:the sum of whose elements is a multiple of 620: 606: 109:{\displaystyle \mathbb {Z} /n\mathbb {Z} } 317: 130: 102: 101: 93: 89: 88: 86: 50:such that every sequence of elements of 359:"A Weighted Erdős-Ginzburg-Ziv Theorem" 260: 553:, Kluwer Acad. Publ. (1993) pp. 19–29 46:, one asks for the smallest value of 7: 574: 572: 170:− 1 integers has a subset of size 81:. They proved that for the group 31:problems about the structure of a 14: 194:-subset summing to a multiple of 576: 198:.) This result is known as the 162:Explicitly this says that any 1: 529:(open-access journal article) 451:Graduate Texts in Mathematics 445:Nathanson, Melvyn B. (1996). 592:. You can help Knowledge by 527:Zero-sum problems - A survey 506:Erdős, Ginzburg, Ziv Theorem 488:"Erdös-Ginzburg-Ziv theorem" 190:− 1 copies of 1 contains no 558:Zero-sum problems: a survey 493:Encyclopedia of Mathematics 357:Grynkiewicz, D. J. (2006), 675: 571: 200:Erdős–Ginzburg–Ziv theorem 378:10.1007/s00493-006-0025-y 328:10.1007/s11139-006-0256-y 270:Bull. Res. Council Israel 204:Cauchy–Davenport theorem 153:{\displaystyle k=2n-1.} 588:-related article is a 249:Zero-sum Ramsey theory 154: 110: 659:Mathematical problems 306:The Ramanujan Journal 294:Nathanson (1996) p.48 155: 111: 27:are certain kinds of 227:David J. Grynkiewicz 223:weighted EGZ theorem 215:Kemnitz's conjecture 186:− 1 copies of 0 and 129: 85: 33:finite abelian group 16:Mathematical problem 639:Combinatorics stubs 535:(workshop homepage) 244:Subset sum problem 239:Davenport constant 221:in 2003), and the 150: 106: 62:terms that sum to 601: 600: 568:(1996) pp. 93–113 453:. Vol. 165. 429:978-3-7643-8961-1 25:zero-sum problems 666: 622: 615: 608: 580: 573: 539:Arie Bialostocki 501: 476: 441: 413: 398: 396: 363: 354: 348: 346: 321: 312:(1–3): 333–337, 301: 295: 292: 286: 285: 265: 219:Christian Reiher 159: 157: 156: 151: 115: 113: 112: 107: 105: 97: 92: 75:Abraham Ginzburg 674: 673: 669: 668: 667: 665: 664: 663: 629: 628: 627: 626: 523: 521:Further reading 486: 483: 465: 455:Springer-Verlag 444: 430: 405: 402: 401: 361: 356: 355: 351: 303: 302: 298: 293: 289: 267: 266: 262: 257: 235: 211:Olson's theorem 127: 126: 83: 82: 39:and a positive 17: 12: 11: 5: 672: 670: 662: 661: 656: 651: 646: 641: 631: 630: 625: 624: 617: 610: 602: 599: 598: 581: 570: 569: 562:Discrete Math. 554: 536: 530: 522: 519: 518: 517: 508: 502: 482: 481:External links 479: 478: 477: 463: 442: 428: 400: 399: 372:(4): 445–453, 349: 296: 287: 259: 258: 256: 253: 252: 251: 246: 241: 234: 231: 149: 146: 143: 140: 137: 134: 104: 100: 96: 91: 15: 13: 10: 9: 6: 4: 3: 2: 671: 660: 657: 655: 652: 650: 649:Combinatorics 647: 645: 644:Ramsey theory 642: 640: 637: 636: 634: 623: 618: 616: 611: 609: 604: 603: 597: 595: 591: 587: 586:combinatorics 582: 579: 575: 567: 563: 559: 555: 552: 551:Nato ASI Ser. 548: 544: 540: 537: 534: 531: 528: 525: 524: 520: 516: 512: 509: 507: 503: 499: 495: 494: 489: 485: 484: 480: 474: 470: 466: 464:0-387-94655-1 460: 456: 452: 448: 443: 439: 435: 431: 425: 421: 417: 412: 411: 404: 403: 395: 391: 387: 383: 379: 375: 371: 367: 366:Combinatorica 360: 353: 350: 345: 341: 337: 333: 329: 325: 320: 315: 311: 307: 300: 297: 291: 288: 283: 279: 275: 271: 264: 261: 254: 250: 247: 245: 242: 240: 237: 236: 232: 230: 228: 224: 220: 216: 212: 207: 205: 201: 197: 193: 189: 185: 181: 177: 173: 169: 165: 160: 147: 144: 141: 138: 135: 132: 124: 122: 119: 98: 94: 80: 76: 72: 67: 65: 61: 57: 53: 49: 45: 42: 38: 34: 30: 29:combinatorial 26: 22: 21:number theory 594:expanding it 583: 565: 561: 550: 546: 511:Sun, Zhi-Wei 491: 446: 416:Solymosi, J. 409: 369: 365: 352: 309: 305: 299: 290: 273: 269: 263: 208: 199: 195: 191: 187: 183: 179: 175: 171: 167: 161: 125: 120: 116:of integers 68: 59: 55: 51: 47: 43: 36: 24: 18: 504:PlanetMath 225:(proved by 217:(proved by 79:Abraham Ziv 654:Paul Erdős 633:Categories 556:Y. Caro, " 473:0859.11003 438:1221.20045 394:1121.11018 344:1126.11011 319:1603.06161 282:0063.00009 255:References 229:in 2005). 71:Paul Erdős 498:EMS Press 336:119600313 276:: 41–43. 145:− 58:contains 386:33448594 233:See also 164:multiset 54:of size 500:, 2001 41:integer 471:  461:  436:  426:  392:  384:  342:  334:  280:  118:modulo 77:, and 584:This 422:–86. 382:S2CID 362:(PDF) 332:S2CID 314:arXiv 590:stub 459:ISBN 424:ISBN 166:of 2 566:152 541:, " 469:Zbl 434:Zbl 390:Zbl 374:doi 340:Zbl 324:doi 278:Zbl 274:10F 213:, 19:In 635:: 564:, 560:" 549:, 513:, 496:, 490:, 467:. 457:. 449:. 432:. 388:, 380:, 370:26 368:, 364:, 338:, 330:, 322:, 310:13 308:, 272:. 206:. 148:1. 123:, 73:, 66:. 23:, 621:e 614:t 607:v 596:. 475:. 440:. 420:1 397:. 376:: 347:. 326:: 316:: 284:. 196:n 192:n 188:n 184:n 180:n 176:n 172:n 168:n 142:n 139:2 136:= 133:k 121:n 103:Z 99:n 95:/ 90:Z 64:0 60:n 56:k 52:G 48:k 44:n 37:G

Index

number theory
combinatorial
finite abelian group
integer
0
Paul Erdős
Abraham Ginzburg
Abraham Ziv
modulo
multiset
Cauchy–Davenport theorem
Olson's theorem
Kemnitz's conjecture
Christian Reiher
weighted EGZ theorem
David J. Grynkiewicz
Davenport constant
Subset sum problem
Zero-sum Ramsey theory
Zbl
0063.00009
arXiv
1603.06161
doi
10.1007/s11139-006-0256-y
S2CID
119600313
Zbl
1126.11011
"A Weighted Erdős-Ginzburg-Ziv Theorem"

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