132:
36:
234:
77:
908:
1145:
584:
318:
equipped with multiple common differences – whereas an arithmetic progression is generated by a single common difference, a generalized arithmetic progression can be generated by multiple common differences. For example, the
686:
1230:
1027:
394:
434:
816:
736:
957:
1288:
1259:
1015:
986:
811:
149:
49:
930:
770:
95:
1168:
746:
of the set can differ from the size if some elements of the set have multiple representations. If the cardinality equals the size, the progression is called
55:
592:
196:
168:
1177:
175:
1379:
288:
270:
252:
215:
182:
113:
63:
164:
1140:{\displaystyle \left\{\mathbf {v} +\sum _{i=1}^{m}k_{i}\mathbf {v} _{i}\,\colon \,k_{1},\dots ,k_{m}\in \mathbb {N} \right\},}
153:
1371:
579:{\displaystyle \{x_{0}+\ell _{1}x_{1}+\cdots +\ell _{d}x_{d}:0\leq \ell _{1}<L_{1},\ldots ,0\leq \ell _{d}<L_{d}\}}
903:{\displaystyle \mathbf {v} ,\mathbf {v} +\mathbf {v} ',\mathbf {v} +2\mathbf {v} ',\mathbf {v} +3\mathbf {v} ',\ldots }
325:
1416:
404:
generalizes this idea to multiple dimensions -- it is a set of vectors of integers, rather than a set of integers.
189:
691:
142:
1411:
750:. Generalized arithmetic progressions can be thought of as a projection of a higher dimensional grid into
315:
1302:
17:
1264:
1235:
991:
962:
787:
1314:
935:
396:
is not an arithmetic progression, but is instead generated by starting with 17 and adding either 3
913:
753:
1295:
1375:
425:
1406:
1385:
1347:
1389:
1153:
1400:
743:
303:
131:
1352:
1335:
988:, called the initial vector and common difference respectively. A subset of
773:
681:{\displaystyle x_{0},x_{1},\dots ,x_{d},L_{1},\dots ,L_{d}\in \mathbb {Z} }
320:
1171:
1225:{\displaystyle \mathbf {v} ,\mathbf {v} _{1},\dots ,\mathbf {v} _{m}}
243:
provides insufficient context for those unfamiliar with the subject
776:
if and only if the generalized arithmetic progression is proper.
1368:
Additive Number Theory: Inverse
Problems and Geometry of Sumsets
400:
5, thus allowing multiple common differences to generate it. A
227:
125:
70:
29:
248:
91:
1301:
The semilinear sets are exactly the sets definable in
1267:
1238:
1180:
1156:
1030:
994:
965:
938:
916:
819:
790:
756:
694:
595:
437:
328:
156:. Unsourced material may be challenged and removed.
86:
may be too technical for most readers to understand
1282:
1253:
1224:
1162:
1139:
1009:
980:
951:
924:
902:
805:
764:
730:
680:
578:
388:
389:{\displaystyle 17,20,22,23,25,26,27,28,29,\dots }
1336:"Semigroups, Presburger Formulas, and Languages"
1334:Ginsburg, Seymour; Spanier, Edwin Henry (1966).
742:of the generalized arithmetic progression; the
8:
573:
438:
64:Learn how and when to remove these messages
1351:
1274:
1270:
1269:
1266:
1245:
1241:
1240:
1237:
1216:
1211:
1195:
1190:
1181:
1179:
1155:
1125:
1124:
1115:
1096:
1091:
1087:
1081:
1076:
1069:
1059:
1048:
1036:
1029:
1001:
997:
996:
993:
972:
968:
967:
964:
940:
937:
917:
915:
885:
873:
861:
849:
837:
828:
820:
818:
797:
793:
792:
789:
758:
757:
755:
722:
709:
699:
693:
674:
673:
664:
645:
632:
613:
600:
594:
567:
554:
529:
516:
497:
487:
468:
458:
445:
436:
414:finite generalized arithmetic progression
408:Finite generalized arithmetic progression
327:
289:Learn how and when to remove this message
271:Learn how and when to remove this message
216:Learn how and when to remove this message
114:Learn how and when to remove this message
98:, without removing the technical details.
418:generalized arithmetic progression (GAP)
1326:
784:Formally, an arithmetic progression of
18:Multidimensional arithmetic progression
731:{\displaystyle L_{1}L_{2}\cdots L_{d}}
253:providing more context for the reader
96:make it understandable to non-experts
7:
813:is an infinite sequence of the form
165:"Generalized arithmetic progression"
154:adding citations to reliable sources
308:generalized arithmetic progression
25:
45:This article has multiple issues.
1283:{\displaystyle \mathbb {N} ^{d}}
1254:{\displaystyle \mathbb {N} ^{d}}
1212:
1191:
1182:
1077:
1037:
1010:{\displaystyle \mathbb {N} ^{d}}
981:{\displaystyle \mathbb {N} ^{d}}
941:
918:
886:
874:
862:
850:
838:
829:
821:
806:{\displaystyle \mathbb {N} ^{d}}
232:
130:
75:
34:
312:multiple arithmetic progression
141:needs additional citations for
53:or discuss these issues on the
1340:Pacific Journal of Mathematics
1:
1372:Graduate Texts in Mathematics
1366:Nathanson, Melvyn B. (1996).
952:{\displaystyle \mathbf {v} '}
925:{\displaystyle \mathbf {v} }
765:{\displaystyle \mathbb {Z} }
314:) is a generalization of an
1374:. Vol. 165. Springer.
1433:
27:Type of numeric sequence
1353:10.2140/pjm.1966.16.285
1284:
1255:
1226:
1164:
1141:
1064:
1011:
982:
953:
926:
904:
807:
766:
732:
682:
580:
390:
316:arithmetic progression
1303:Presburger arithmetic
1285:
1256:
1232:are fixed vectors in
1227:
1165:
1142:
1044:
1021:if it is of the form
1012:
983:
959:are fixed vectors in
954:
927:
905:
808:
772:. This projection is
767:
733:
683:
581:
391:
1265:
1236:
1178:
1154:
1028:
992:
963:
936:
914:
817:
788:
754:
692:
593:
435:
416:, or sometimes just
326:
150:improve this article
424:is defined to be a
249:improve the article
1294:if it is a finite
1280:
1251:
1222:
1160:
1137:
1007:
978:
949:
922:
900:
803:
762:
728:
678:
576:
386:
1417:Arithmetic series
1315:Freiman's theorem
1163:{\displaystyle m}
299:
298:
291:
281:
280:
273:
226:
225:
218:
200:
124:
123:
116:
68:
16:(Redirected from
1424:
1393:
1358:
1357:
1355:
1331:
1298:of linear sets.
1289:
1287:
1286:
1281:
1279:
1278:
1273:
1260:
1258:
1257:
1252:
1250:
1249:
1244:
1231:
1229:
1228:
1223:
1221:
1220:
1215:
1200:
1199:
1194:
1185:
1169:
1167:
1166:
1161:
1146:
1144:
1143:
1138:
1133:
1129:
1128:
1120:
1119:
1101:
1100:
1086:
1085:
1080:
1074:
1073:
1063:
1058:
1040:
1016:
1014:
1013:
1008:
1006:
1005:
1000:
987:
985:
984:
979:
977:
976:
971:
958:
956:
955:
950:
948:
944:
931:
929:
928:
923:
921:
909:
907:
906:
901:
893:
889:
877:
869:
865:
853:
845:
841:
832:
824:
812:
810:
809:
804:
802:
801:
796:
771:
769:
768:
763:
761:
737:
735:
734:
729:
727:
726:
714:
713:
704:
703:
687:
685:
684:
679:
677:
669:
668:
650:
649:
637:
636:
618:
617:
605:
604:
585:
583:
582:
577:
572:
571:
559:
558:
534:
533:
521:
520:
502:
501:
492:
491:
473:
472:
463:
462:
450:
449:
395:
393:
392:
387:
294:
287:
276:
269:
265:
262:
256:
236:
235:
228:
221:
214:
210:
207:
201:
199:
158:
134:
126:
119:
112:
108:
105:
99:
79:
78:
71:
60:
38:
37:
30:
21:
1432:
1431:
1427:
1426:
1425:
1423:
1422:
1421:
1397:
1396:
1382:
1365:
1362:
1361:
1333:
1332:
1328:
1323:
1311:
1268:
1263:
1262:
1239:
1234:
1233:
1210:
1189:
1176:
1175:
1152:
1151:
1111:
1092:
1075:
1065:
1035:
1031:
1026:
1025:
995:
990:
989:
966:
961:
960:
939:
934:
933:
912:
911:
884:
860:
836:
815:
814:
791:
786:
785:
782:
780:Semilinear sets
752:
751:
718:
705:
695:
690:
689:
660:
641:
628:
609:
596:
591:
590:
563:
550:
525:
512:
493:
483:
464:
454:
441:
433:
432:
420:, of dimension
410:
324:
323:
301:
295:
284:
283:
282:
277:
266:
260:
257:
246:
237:
233:
222:
211:
205:
202:
159:
157:
147:
135:
120:
109:
103:
100:
92:help improve it
89:
80:
76:
39:
35:
28:
23:
22:
15:
12:
11:
5:
1430:
1428:
1420:
1419:
1414:
1409:
1399:
1398:
1395:
1394:
1380:
1360:
1359:
1346:(2): 285–296.
1325:
1324:
1322:
1319:
1318:
1317:
1310:
1307:
1290:is said to be
1277:
1272:
1261:. A subset of
1248:
1243:
1219:
1214:
1209:
1206:
1203:
1198:
1193:
1188:
1184:
1159:
1148:
1147:
1136:
1132:
1127:
1123:
1118:
1114:
1110:
1107:
1104:
1099:
1095:
1090:
1084:
1079:
1072:
1068:
1062:
1057:
1054:
1051:
1047:
1043:
1039:
1034:
1017:is said to be
1004:
999:
975:
970:
947:
943:
920:
899:
896:
892:
888:
883:
880:
876:
872:
868:
864:
859:
856:
852:
848:
844:
840:
835:
831:
827:
823:
800:
795:
781:
778:
760:
738:is called the
725:
721:
717:
712:
708:
702:
698:
688:. The product
676:
672:
667:
663:
659:
656:
653:
648:
644:
640:
635:
631:
627:
624:
621:
616:
612:
608:
603:
599:
587:
586:
575:
570:
566:
562:
557:
553:
549:
546:
543:
540:
537:
532:
528:
524:
519:
515:
511:
508:
505:
500:
496:
490:
486:
482:
479:
476:
471:
467:
461:
457:
453:
448:
444:
440:
409:
406:
402:semilinear set
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
297:
296:
279:
278:
240:
238:
231:
224:
223:
138:
136:
129:
122:
121:
83:
81:
74:
69:
43:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1429:
1418:
1415:
1413:
1412:Combinatorics
1410:
1408:
1405:
1404:
1402:
1391:
1387:
1383:
1381:0-387-94655-1
1377:
1373:
1369:
1364:
1363:
1354:
1349:
1345:
1341:
1337:
1330:
1327:
1320:
1316:
1313:
1312:
1308:
1306:
1304:
1299:
1297:
1293:
1275:
1246:
1217:
1207:
1204:
1201:
1196:
1186:
1173:
1157:
1134:
1130:
1121:
1116:
1112:
1108:
1105:
1102:
1097:
1093:
1088:
1082:
1070:
1066:
1060:
1055:
1052:
1049:
1045:
1041:
1032:
1024:
1023:
1022:
1020:
1002:
973:
945:
897:
894:
890:
881:
878:
870:
866:
857:
854:
846:
842:
833:
825:
798:
779:
777:
775:
749:
745:
741:
723:
719:
715:
710:
706:
700:
696:
670:
665:
661:
657:
654:
651:
646:
642:
638:
633:
629:
625:
622:
619:
614:
610:
606:
601:
597:
568:
564:
560:
555:
551:
547:
544:
541:
538:
535:
530:
526:
522:
517:
513:
509:
506:
503:
498:
494:
488:
484:
480:
477:
474:
469:
465:
459:
455:
451:
446:
442:
431:
430:
429:
427:
423:
419:
415:
407:
405:
403:
399:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
322:
317:
313:
309:
305:
293:
290:
275:
272:
264:
254:
250:
244:
241:This article
239:
230:
229:
220:
217:
209:
198:
195:
191:
188:
184:
181:
177:
174:
170:
167: –
166:
162:
161:Find sources:
155:
151:
145:
144:
139:This article
137:
133:
128:
127:
118:
115:
107:
97:
93:
87:
84:This article
82:
73:
72:
67:
65:
58:
57:
52:
51:
46:
41:
32:
31:
19:
1367:
1343:
1339:
1329:
1300:
1291:
1149:
1018:
783:
747:
739:
588:
428:of the form
421:
417:
413:
411:
401:
397:
311:
307:
300:
285:
267:
258:
247:Please help
242:
212:
203:
193:
186:
179:
172:
160:
148:Please help
143:verification
140:
110:
101:
85:
61:
54:
48:
47:Please help
44:
744:cardinality
304:mathematics
1401:Categories
1390:0859.11003
1321:References
1292:semilinear
176:newspapers
50:improve it
1205:…
1122:∈
1106:…
1089::
1046:∑
898:…
774:injective
716:⋯
671:∈
655:…
623:…
552:ℓ
548:≤
539:…
514:ℓ
510:≤
485:ℓ
478:⋯
456:ℓ
384:…
261:June 2024
206:June 2024
104:June 2024
56:talk page
1309:See also
1170:is some
946:′
910:, where
891:′
867:′
843:′
321:sequence
1407:Algebra
1172:integer
190:scholar
90:Please
1388:
1378:
1150:where
1019:linear
748:proper
589:where
192:
185:
178:
171:
163:
1296:union
197:JSTOR
183:books
1376:ISBN
1174:and
932:and
740:size
561:<
523:<
310:(or
306:, a
169:news
1386:Zbl
1348:doi
426:set
302:In
251:by
152:by
94:to
1403::
1384:.
1370:.
1344:16
1342:.
1338:.
1305:.
412:A
398:or
378:29
372:28
366:27
360:26
354:25
348:23
342:22
336:20
330:17
59:.
1392:.
1356:.
1350::
1276:d
1271:N
1247:d
1242:N
1218:m
1213:v
1208:,
1202:,
1197:1
1192:v
1187:,
1183:v
1158:m
1135:,
1131:}
1126:N
1117:m
1113:k
1109:,
1103:,
1098:1
1094:k
1083:i
1078:v
1071:i
1067:k
1061:m
1056:1
1053:=
1050:i
1042:+
1038:v
1033:{
1003:d
998:N
974:d
969:N
942:v
919:v
895:,
887:v
882:3
879:+
875:v
871:,
863:v
858:2
855:+
851:v
847:,
839:v
834:+
830:v
826:,
822:v
799:d
794:N
759:Z
724:d
720:L
711:2
707:L
701:1
697:L
675:Z
666:d
662:L
658:,
652:,
647:1
643:L
639:,
634:d
630:x
626:,
620:,
615:1
611:x
607:,
602:0
598:x
574:}
569:d
565:L
556:d
545:0
542:,
536:,
531:1
527:L
518:1
507:0
504::
499:d
495:x
489:d
481:+
475:+
470:1
466:x
460:1
452:+
447:0
443:x
439:{
422:d
381:,
375:,
369:,
363:,
357:,
351:,
345:,
339:,
333:,
292:)
286:(
274:)
268:(
263:)
259:(
255:.
245:.
219:)
213:(
208:)
204:(
194:·
187:·
180:·
173:·
146:.
117:)
111:(
106:)
102:(
88:.
66:)
62:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
