118:
212:
511:
544:
756:
49:
814:
542:
J.W.S. Cassels; H.P.F. Swinnerton-Dyer (1955-06-23). "On the product of three homogeneous linear forms and the indefinite ternary quadratic forms".
126:
725:
804:
262:-coordinate. This product will certainly be at most 1/4. The conjecture makes no statement about whether this sequence of values will
428:
267:
620:
M. Einsiedler; A. Katok; E. Lindenstrauss (2006-09-01). "Invariant measures and the set of exceptions to
Littlewood's conjecture".
445:
310:. This can be formulated another way, in group-theoretic terms. There is now another conjecture, expected to hold for
442:
These results imply that non-trivial pairs satisfying the conjecture exist: indeed, given a real number α such that
717:
403:
Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is of
263:
32:
36:
432:
622:
522:
302:
point of a product of three linear forms in three real variables: the implication was shown in 1955 by
270:, and says that there is a subsequence for which the distances decay faster than the reciprocal, i.e.
809:
709:
641:
553:
420:
295:
665:
631:
585:
569:
416:
408:
373:
708:
Adamczewski, Boris; Bugeaud, Yann (2010). "8. Transcendence and diophantine approximation". In
721:
345:
783:
765:
731:
673:
649:
593:
561:
404:
388:
303:
779:
661:
581:
787:
775:
735:
677:
657:
597:
577:
307:
299:
645:
557:
113:{\displaystyle \liminf _{n\to \infty }\ n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0,}
283:
513:, it is possible to construct an explicit β such that (α,β) satisfies the conjecture.
798:
589:
424:
412:
28:
770:
751:
669:
653:
40:
20:
752:"The work of Einsiedler, Katok, and Lindenstrauss on the Littlewood conjecture"
716:. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge:
207:{\displaystyle \Vert x\Vert :=\min(|x-\lfloor x\rfloor |,|x-\lceil x\rceil |)}
392:
266:; it typically does not, in fact. The conjecture states something about the
565:
254:
For each of these, multiply the distance to the closest line with integer
636:
573:
258:-coordinate by the distance to the closest line with integer
387:
This in turn is a special case of a general conjecture of
435:
for diagonalizable actions of higher-rank groups, and an
230:) in the plane, and then consider the sequence of points
294:
It is known that this would follow from a result in the
506:{\displaystyle \inf _{n\geq 1}n\cdot ||n\alpha ||>0}
448:
129:
52:
505:
206:
112:
545:Philosophical Transactions of the Royal Society A
450:
142:
54:
431:zero. The result was proved by using a measure
423:zero; and in fact is a union of countably many
8:
193:
187:
165:
159:
136:
130:
98:
89:
85:
76:
714:Combinatorics, automata, and number theory
769:
635:
492:
487:
476:
471:
453:
447:
439:proved by Lindenstrauss and Barak Weiss.
196:
176:
168:
148:
128:
88:
75:
57:
51:
222:This means the following: take a point (
214:is the distance to the nearest integer.
39:around 1930. It states that for any two
534:
699:Adamczewski & Bugeaud (2010) p.446
690:Adamczewski & Bugeaud (2010) p.445
610:Adamczewski & Bugeaud (2010) p.444
7:
298:, about the minimum on a non-zero
64:
14:
290:Connection to further conjectures
815:Unsolved problems in mathematics
750:Akshay Venkatesh (2007-10-29).
493:
488:
477:
472:
314:≥ 3: it is stated in terms of
201:
197:
177:
169:
149:
145:
61:
1:
771:10.1090/S0273-0979-07-01194-9
757:Bull. Amer. Math. Soc. (N.S.)
419:have shown that it must have
654:10.4007/annals.2006.164.513
218:Formulation and explanation
831:
718:Cambridge University Press
805:Diophantine approximation
33:Diophantine approximation
712:; Rigo, Michael (eds.).
37:John Edensor Littlewood
566:10.1098/rsta.1955.0010
507:
433:classification theorem
429:box-counting dimension
208:
114:
31:(as of April 2024) in
623:Annals of Mathematics
523:Littlewood polynomial
508:
209:
115:
25:Littlewood conjecture
720:. pp. 410–451.
446:
340:), and the subgroup
127:
50:
16:Mathematical problem
646:2006math.....12721E
558:1955RSPTA.248...73C
421:Hausdorff dimension
296:geometry of numbers
503:
464:
417:Elon Lindenstrauss
409:Manfred Einsiedler
374:relatively compact
204:
110:
68:
727:978-0-521-51597-9
449:
437:isolation theorem
346:diagonal matrices
284:little-o notation
71:
53:
822:
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539:
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405:Lebesgue measure
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744:Further reading
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710:Berthé, Valérie
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637:math.DS/0612721
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401:
399:Partial results
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308:Swinnerton-Dyer
292:
220:
125:
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47:
17:
12:
11:
5:
828:
826:
818:
817:
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797:
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793:
792:
764:(1): 117–134.
745:
742:
741:
740:
726:
702:
701:
692:
683:
630:(2): 513–560.
612:
603:
552:(940): 73–96.
533:
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332:
321:
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268:limit inferior
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63:
60:
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55:lim inf
35:, proposed by
15:
13:
10:
9:
6:
4:
3:
2:
827:
816:
813:
811:
808:
806:
803:
802:
800:
789:
785:
781:
777:
772:
767:
763:
759:
758:
753:
748:
747:
743:
737:
733:
729:
723:
719:
715:
711:
706:
705:
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693:
687:
684:
679:
675:
671:
667:
663:
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647:
643:
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633:
629:
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607:
604:
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587:
583:
579:
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559:
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481:
468:
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438:
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422:
418:
414:
413:Anatole Katok
410:
406:
398:
396:
394:
390:
385:
383:
379:
375:
371:
368:/Γ such that
367:
363:
359:
358:
353:
351:
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335:
328:
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107:
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58:
46:
45:
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38:
34:
30:
26:
22:
761:
755:
713:
695:
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627:
621:
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606:
549:
543:
537:
441:
436:
425:compact sets
402:
386:
381:
377:
369:
365:
361:
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355:
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349:
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247:
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122:
41:real numbers
29:open problem
24:
18:
810:Conjectures
384:is closed.
21:mathematics
799:Categories
788:1194.11075
736:1271.11073
678:1109.22004
598:0065.27905
529:References
393:Lie groups
380:/Γ), then
360:: for any
357:Conjecture
590:122708867
485:α
469:⋅
458:≥
194:⌉
188:⌈
185:−
166:⌋
160:⌊
157:−
137:‖
131:‖
99:‖
96:β
90:‖
86:‖
83:α
77:‖
65:∞
62:→
43:α and β,
517:See also
389:Margulis
264:converge
250:), ... .
780:2358379
662:2247967
642:Bibcode
582:0070653
554:Bibcode
329:), Γ =
304:Cassels
300:lattice
282:in the
786:
778:
734:
724:
676:
670:613883
668:
660:
596:
588:
580:
572:
407:zero.
123:where
70:
27:is an
23:, the
666:S2CID
632:arXiv
586:S2CID
574:91633
570:JSTOR
242:), (3
722:ISBN
498:>
415:and
376:(in
306:and
274:o(1/
784:Zbl
766:doi
732:Zbl
674:Zbl
650:doi
628:164
594:Zbl
562:doi
550:248
451:inf
427:of
391:on
372:is
364:in
348:in
344:of
246:, 3
238:, 2
143:min
19:In
801::
782:.
776:MR
774:.
762:45
760:.
754:.
730:.
672:.
664:.
658:MR
656:.
648:.
640:.
626:.
592:.
584:.
578:MR
576:.
568:.
560:.
548:.
411:,
395:.
382:Dg
370:Dg
352:.
331:SL
320:SL
318:=
286:.
234:(2
226:,
140::=
790:.
768::
738:.
680:.
652::
644::
634::
600:.
564::
556::
501:0
494:|
489:|
482:n
478:|
473:|
466:n
461:1
455:n
378:G
366:G
362:g
350:G
342:D
338:Z
336:(
333:n
327:R
325:(
322:n
316:G
312:n
278:)
276:n
260:y
256:x
248:β
244:α
240:β
236:α
228:β
224:α
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198:|
191:x
182:x
178:|
174:,
170:|
163:x
154:x
150:|
146:(
134:x
108:,
105:0
102:=
93:n
80:n
73:n
59:n
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