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Beresnevich, Victor; Velani, Sanju (2006). "A mass transference principle and the Duffin-Schaeffer conjecture for
Hausdorff measures".
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is the crux of the conjecture. There have been many partial results of the Duffin–Schaeffer conjecture established to date.
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420:
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59:
114:
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to the original Duffin–Schaeffer conjecture, which is a priori weaker. This result was published in the
735:{\displaystyle \sum _{n=1}^{\infty }\left({\frac {f(n)}{n}}\right)^{1+\varepsilon }\varphi (n)=\infty .}
615:
1243:
753:
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416:
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Duffin, R. J.; Schaeffer, A. C. (1941). "Khintchine's problem in metric diophantine approximation".
561:
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873:
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399:
A higher-dimensional analogue of this conjecture was resolved by
Vaughan and Pollington in 1990.
366:
55:
612:. More recently, this was strengthened to the conjecture being true whenever there exists some
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Ten lectures on the interface between analytic number theory and harmonic analysis
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Surveys in number theory: Papers from the millennial conference on number theory
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39:
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1152:. London Mathematical Society Monographs. New Series. Vol. 18. Oxford:
903:. Regional Conference Series in Mathematics. Vol. 84. Providence, RI:
353:{\displaystyle \sum _{q=1}^{\infty }\varphi (q){\frac {f(q)}{q}}=\infty ,}
431:
established in 1970 that the conjecture holds if there exists a constant
210:{\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {f(q)}{q}}}
458:
35:
1097:
852:
1182:(2002). "One hundred years of normal numbers". In Bennett, M. A.;
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558:. This was strengthened by Jeffrey Vaaler in 1978 to the case
1224:
Quanta magazine article about Duffin-Schaeffer conjecture.
102:{\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{+}}
1229:
Numberphile interview with James
Maynard about the proof.
1190:; Diamond, H.G.; Hildebrand, A.J.; Philipp, W. (eds.).
34:, also known as the Duffin-Schaeffer conjecture, is a
1066:
The Duffin-Schaeffer
Conjecture with extra divergence
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834:Koukoulopoulos, Dimitris; Maynard, James (2020).
745:This was done by Haynes, Pollington, and Velani.
1036:Ohio State University Department of Mathematics
752:analogue of the Duffin–Schaeffer conjecture is
748:In 2006, Beresnevich and Velani proved that a
8:
1194:. Natick, MA: A K Peters. pp. 57–74.
1096:
1064:A. Haynes, A. Pollington, and S. Velani,
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946:dimensional Duffin–Schaeffer conjecture"
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7:
836:"On the Duffin-Schaeffer conjecture"
117:taking on positive values, then for
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344:
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25:
1249:Conjectures that have been proved
631:{\displaystyle \varepsilon >0}
220:has infinitely many solutions in
1070:https://arxiv.org/abs/0811.1234
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605:{\displaystyle f(n)=O(n^{-1})}
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32:Koukoulopoulos–Maynard theorem
1:
1029:"Duffin-Schaeffer Conjecture"
905:American Mathematical Society
805:10.1215/S0012-7094-41-00818-9
46:proposed as a conjecture by
1107:10.4007/annals.2006.164.971
870:10.4007/annals.2020.192.1.5
862:10.4007/annals.2020.192.1.5
385:{\displaystyle \varphi (q)}
18:Duffin–Schaeffer conjecture
1270:
1254:Diophantine approximation
964:10.1112/s0025579300012900
44:Diophantine approximation
1055:Harman (1998) p. 28
1018:Harman (1998) p. 27
1009:Harman (2002) p. 68
1000:Harman (2002) p. 69
516:{\displaystyle f(n)=c/n}
394:Euler's totient function
411:approximations implies
133:{\displaystyle \alpha }
60:Dimitris Koukoulopoulos
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551:{\displaystyle f(n)=0}
517:
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450:{\displaystyle c>0}
407:That existence of the
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268:{\displaystyle q>0}
243:
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1148:Harman, Glyn (1998).
1084:Annals of Mathematics
840:Annals of Mathematics
759:Annals of Mathematics
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638:such that the series
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66:. It states that if
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457:such that for every
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425:converse implication
421:Borel–Cantelli lemma
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42:, specifically, the
27:Mathematical theorem
897:Montgomery, Hugh L.
242:{\displaystyle p,q}
936:Pollington, A.D.;
772:Khinchin's theorem
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1201:978-1-56881-162-8
1163:978-0-19-850083-4
1087:. Second Series.
1068:, arXiv, (2009),
914:978-0-8218-0737-8
750:Hausdorff measure
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473:{\displaystyle n}
419:follows from the
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140:(with respect to
16:(Redirected from
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907:. p. 204.
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1180:Harman, Glyn
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1098:math/0412141
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1040:. Retrieved
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403:Introduction
398:
362:
219:
54:in 1941 and
48:R. J. Duffin
31:
29:
1244:Conjectures
951:Mathematika
58:in 2019 by
40:mathematics
1238:Categories
1210:1062.11052
1188:Boston, N.
1172:1081.11057
1142:References
1131:1148.11033
1042:2019-09-19
988:0715.11036
923:0814.11001
853:1907.04593
846:(1): 251.
821:0025.11002
813:67.0145.03
754:equivalent
429:Paul Erdős
413:divergence
146:inequality
119:almost all
1115:0003-486X
980:122789762
972:0025-5793
878:195874052
727:∞
712:φ
707:ε
665:∞
650:∑
620:ε
592:−
371:φ
345:∞
309:φ
304:∞
289:∑
166:−
163:α
128:α
85:→
1123:14475449
940:(1990).
899:(1994).
766:See also
409:rational
115:function
113:-valued
459:integer
423:. The
415:of the
144:), the
36:theorem
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417:series
363:where
56:proven
1119:S2CID
1093:arXiv
1032:(PDF)
976:S2CID
942:"The
874:S2CID
866:JSTOR
848:arXiv
778:Notes
249:with
109:is a
1196:ISBN
1158:ISBN
1111:ISSN
968:ISSN
909:ISBN
623:>
442:>
260:>
184:<
111:real
62:and
50:and
30:The
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1168:Zbl
1127:Zbl
1103:doi
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984:Zbl
960:doi
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817:Zbl
809:JFM
801:doi
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392:is
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