20:
570:
415:
166:
111:
430:
282:
717:
674:
949:
944:
899:
424:
and any
Lebesgue integrable function Ć’. In modern formulations, it is asked under what conditions the identity
128:
269:
subset of the unit interval. The corresponding generalizations for the Weyl and
Vinogradov results were proven by
35:
are due to the closeness of its value to the rational number 355/113. Similarly, the 7 distinct groups are due to
664:
669:
565:{\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}f((x+b_{k}a){\bmod {1}})=\int _{0}^{1}f(y)\,dy}
84:
75:
939:
237:
181:
615:, there exists a function Ć’ for which the sum diverges. In this sense, this sequence is considered to be a
410:{\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}f((x+ka){\bmod {1}})=\int _{0}^{1}f(y)\,dy}
888:
Ergodic Theory and its
Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference
904:(An extensive survey of the ergodic properties of generalizations of the equidistribution theorem of
807:
41:
640:
266:
864:
782:
734:
247:
895:
835:
679:
118:
19:
856:
825:
815:
774:
726:
652:
643:, which shows that equidistribution is equivalent to having a non-trivial estimate for the
644:
243:
204:
122:
811:
916:
830:
933:
909:
868:
786:
738:
270:
233:
177:
594: mod 1 is uniformly distributed for almost all, but not all, irrational
27:
terms using the equidistribution theorem with four common irrational numbers, for
16:
Integer multiples of any irrational mod 1 are uniformly distributed on the circle
48:
259:
185:
905:
795:
839:
820:
23:
Illustration of filling the unit interval (horizontal axis) with the first
576:
860:
847:
Ya. Khinchin, A. (1933). "Zur
Birkhoff's Lösung des Ergodensproblems".
778:
730:
240:, that every sufficiently large odd number is the sum of three primes.
647:
formed with the sequence as exponents. For the case of multiples of
203:, ... mod 1 is uniformly distributed on the unit interval. In 1937,
79:
762:
712:
698:Über ein in der Theorie der säkularen Störungen vorkommendes Problem
713:"Über die Gibbs'sche Erscheinung und verwandte Konvergenzphänomene"
188:, variants of this theorem continue to be studied to this day.
513:
358:
176:
While this theorem was proved in 1909 and 1910 separately by
31:
from 0 to 999 (vertical axis). The 113 distinct bands for
651:, Weyl's criterion reduces the problem to summing finite
925:(Proof of the Weyl's theorem based on Fourier Analysis)
433:
285:
131:
87:
636:, because it does not have the latter shortcoming.
751:Bull Intl. Acad. Polonaise des Sci. et des Lettres
564:
409:
160:
105:
890:, (1995) Karl E. Petersen and Ibrahim A. Salama,
763:"Ueber die Gleichverteilung von Zahlen mod. Eins"
923:, (2003) Princeton University Press, pp 105–113
884:Pointwise ergodic theorems via harmonic analysis
435:
287:
276:Specifically, Khinchin showed that the identity
747:Sur la valeur asymptotique d'une certaine somme
161:{\displaystyle \mu ={\frac {d\theta }{2\pi }}}
236:. Vinogradov's proof was a byproduct of the
125:where one takes the normalized angle measure
8:
912:. Focuses on methods developed by Bourgain.)
718:Rendiconti del Circolo Matematico di Palermo
590:One noteworthy result is that the sequence 2
250:, in 1933, proved that the generalization
894:, Cambridge University Press, Cambridge,
829:
819:
555:
537:
532:
516:
512:
500:
475:
464:
450:
438:
432:
400:
382:
377:
361:
357:
327:
316:
302:
290:
284:
138:
130:
106:{\displaystyle \mathbb {R} /\mathbb {Z} }
99:
98:
93:
89:
88:
86:
18:
191:In 1916, Weyl proved that the sequence
882:Joseph M. Rosenblatt and Máté Weirdl,
607: = 2a, for every irrational
219:mod 1 is uniformly distributed, where
7:
634:universally good averaging sequence
55:is the statement that the sequence
617:universally bad averaging sequence
445:
297:
14:
921:Fourier Analysis. An Introduction
675:Dirichlet's approximation theorem
575:might hold, given some general
121:. It is a special case of the
796:"Proof of the ergodic theorem"
598:. Similarly, for the sequence
552:
546:
522:
509:
487:
484:
442:
397:
391:
367:
354:
339:
336:
294:
1:
639:A powerful general result is
265:, is equidistributed on any
800:Proc. Natl. Acad. Sci. U.S.A
966:
950:Theorems in number theory
945:Diophantine approximation
665:Diophantine approximation
207:proved that the sequence
42:(click for detailed view)
39:being approximately 22/7.
794:Birkhoff, G. D. (1931).
670:Low-discrepancy sequence
53:equidistribution theorem
238:odd Goldbach conjecture
886:, (1993) appearing in
821:10.1073/pnas.17.12.656
745:W. Sierpinski, (1910)
566:
480:
411:
332:
162:
107:
44:
702:J. reine angew. Math.
691:Historical references
567:
460:
420:holds for almost all
412:
312:
163:
108:
76:uniformly distributed
22:
919:and Rami Shakarchi,
632:, which is termed a
431:
283:
129:
85:
812:1931PNAS...17..656B
542:
387:
267:Lebesgue measurable
861:10.1007/BF01448905
779:10.1007/BF01475864
731:10.1007/bf03014883
562:
528:
449:
407:
373:
301:
248:Aleksandr Khinchin
158:
103:
45:
877:Modern references
761:Weyl, H. (1916).
711:Weyl, H. (1910).
680:Three-gap theorem
611:, and almost all
458:
434:
310:
286:
182:Wacław Sierpiński
156:
119:irrational number
957:
872:
843:
833:
823:
790:
742:
696:P. Bohl, (1909)
653:geometric series
645:exponential sums
641:Weyl's criterion
619:, as opposed to
571:
569:
568:
563:
541:
536:
521:
520:
505:
504:
479:
474:
459:
451:
448:
416:
414:
413:
408:
386:
381:
366:
365:
331:
326:
311:
303:
300:
167:
165:
164:
159:
157:
155:
147:
139:
112:
110:
109:
104:
102:
97:
92:
38:
34:
965:
964:
960:
959:
958:
956:
955:
954:
930:
929:
879:
846:
806:(12): 656–660.
793:
760:
710:
693:
688:
661:
627:
606:
586:
496:
429:
428:
281:
280:
246:, in 1931, and
244:George Birkhoff
227:
215:
205:Ivan Vinogradov
174:
148:
140:
127:
126:
123:ergodic theorem
83:
82:
40:
36:
32:
17:
12:
11:
5:
963:
961:
953:
952:
947:
942:
940:Ergodic theory
932:
931:
928:
927:
917:Elias M. Stein
914:
878:
875:
874:
873:
844:
791:
773:(3): 313–352.
758:
743:
708:
707:, pp. 189–283.
692:
689:
687:
684:
683:
682:
677:
672:
667:
660:
657:
623:
602:
582:
573:
572:
561:
558:
554:
551:
548:
545:
540:
535:
531:
527:
524:
519:
515:
511:
508:
503:
499:
495:
492:
489:
486:
483:
478:
473:
470:
467:
463:
457:
454:
447:
444:
441:
437:
418:
417:
406:
403:
399:
396:
393:
390:
385:
380:
376:
372:
369:
364:
360:
356:
353:
350:
347:
344:
341:
338:
335:
330:
325:
322:
319:
315:
309:
306:
299:
296:
293:
289:
223:
211:
173:
170:
154:
151:
146:
143:
137:
134:
101:
96:
91:
72:
71:
15:
13:
10:
9:
6:
4:
3:
2:
962:
951:
948:
946:
943:
941:
938:
937:
935:
926:
922:
918:
915:
913:
911:
910:unit interval
907:
901:
900:0-521-45999-0
897:
893:
889:
885:
881:
880:
876:
870:
866:
862:
858:
854:
850:
845:
841:
837:
832:
827:
822:
817:
813:
809:
805:
801:
797:
792:
788:
784:
780:
776:
772:
768:
764:
759:
756:
752:
748:
744:
740:
736:
732:
728:
724:
720:
719:
714:
709:
706:
703:
699:
695:
694:
690:
685:
681:
678:
676:
673:
671:
668:
666:
663:
662:
658:
656:
654:
650:
646:
642:
637:
635:
631:
628: =
626:
622:
618:
614:
610:
605:
601:
597:
593:
588:
585:
581:
578:
559:
556:
549:
543:
538:
533:
529:
525:
517:
506:
501:
497:
493:
490:
481:
476:
471:
468:
465:
461:
455:
452:
439:
427:
426:
425:
423:
404:
401:
394:
388:
383:
378:
374:
370:
362:
351:
348:
345:
342:
333:
328:
323:
320:
317:
313:
307:
304:
291:
279:
278:
277:
274:
272:
271:Jean Bourgain
268:
264:
261:
257:
254: +
253:
249:
245:
241:
239:
235:
231:
226:
222:
218:
214:
210:
206:
202:
198:
194:
189:
187:
183:
179:
171:
169:
152:
149:
144:
141:
135:
132:
124:
120:
116:
94:
81:
77:
69:
65:
61:
58:
57:
56:
54:
50:
43:
30:
26:
21:
924:
920:
903:
891:
887:
883:
852:
848:
803:
799:
770:
766:
754:
750:
746:
722:
716:
704:
701:
697:
648:
638:
633:
629:
624:
620:
616:
612:
608:
603:
599:
595:
591:
589:
583:
579:
574:
421:
419:
275:
262:
255:
251:
242:
229:
224:
220:
216:
212:
208:
200:
196:
192:
190:
178:Hermann Weyl
175:
114:
73:
67:
63:
59:
52:
46:
28:
24:
855:: 485–488.
757:, pp. 9–11.
753:(Cracovie)
725:: 377–407.
70:, ... mod 1
49:mathematics
934:Categories
906:shift maps
686:References
260:almost all
186:Piers Bohl
869:122289068
849:Math. Ann
787:123470919
767:Math. Ann
739:122545523
530:∫
462:∑
446:∞
443:→
375:∫
314:∑
298:∞
295:→
273:in 1988.
153:π
145:θ
133:μ
840:16577406
755:series A
659:See also
577:sequence
908:on the
831:1076138
808:Bibcode
228:is the
172:History
113:, when
78:on the
898:
867:
838:
828:
785:
737:
258:, for
117:is an
80:circle
51:, the
865:S2CID
783:S2CID
735:S2CID
234:prime
896:ISBN
892:eds.
836:PMID
184:and
857:doi
853:107
826:PMC
816:doi
775:doi
727:doi
723:330
705:135
514:mod
436:lim
359:mod
288:lim
232:th
199:, 3
195:, 2
74:is
66:, 3
62:, 2
47:In
936::
902:.
863:.
851:.
834:.
824:.
814:.
804:17
802:.
798:.
781:.
771:77
769:.
765:.
749:,
733:.
721:.
715:.
700:,
655:.
587:.
256:na
180:,
168:.
871:.
859::
842:.
818::
810::
789:.
777::
741:.
729::
649:a
630:k
625:k
621:b
613:x
609:a
604:k
600:b
596:a
592:a
584:k
580:b
560:y
557:d
553:)
550:y
547:(
544:f
539:1
534:0
526:=
523:)
518:1
510:)
507:a
502:k
498:b
494:+
491:x
488:(
485:(
482:f
477:n
472:1
469:=
466:k
456:n
453:1
440:n
422:x
405:y
402:d
398:)
395:y
392:(
389:f
384:1
379:0
371:=
368:)
363:1
355:)
352:a
349:k
346:+
343:x
340:(
337:(
334:f
329:n
324:1
321:=
318:k
308:n
305:1
292:n
263:x
252:x
230:n
225:n
221:p
217:a
213:n
209:p
201:a
197:a
193:a
150:2
142:d
136:=
115:a
100:Z
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90:R
68:a
64:a
60:a
37:Ď€
33:Ď€
29:n
25:n
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