1762:
2740:
2460:
1507:
1285:
548:
Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving no gaps. For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates
527:
585:
of points chosen from a fine partition of the interval. Therefore, if some sequence is equidistributed in , it is expected that this sequence can be used to calculate the integral of a
Riemann-integrable function. This leads to the following criterion for an equidistributed sequence:
2735:{\displaystyle \left|\sum _{j=0}^{n-1}e^{2\pi i\ell j\alpha }\right|=\left|\sum _{j=0}^{n-1}\left(e^{2\pi i\ell \alpha }\right)^{j}\right|=\left|{\frac {1-e^{2\pi i\ell n\alpha }}{1-e^{2\pi i\ell \alpha }}}\right|\leq {\frac {2}{\left|1-e^{2\pi i\ell \alpha }\right|}},}
793:
1291:
1069:
3804:
255:
549:
multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε. For stronger criteria and for constructions of sequences that are more evenly distributed, see
2341:
2051:
1061:
299:
members of the sequence which fall between 0.5 and 0.9 must approach 1/5. Loosely speaking, one could say that each member of the sequence is equally likely to fall anywhere in its range. However, this is not to say that
2173:
from above and below by two continuous functions on the interval, whose integrals differ by an arbitrary ε. By an argument similar to the proof of the
Riemann integral criterion, it is possible to extend the result to any
4016:
360:
3124:
1583:
3390:
881:
being an indicator function of an interval. It remains to assume that the integral criterion holds for indicator functions and prove that it holds for general
Riemann-integrable functions as well.
2134:
658:
874:
3312:
1502:{\displaystyle {\frac {1}{b-a}}\int _{a}^{b}f_{2}(x)\,dx=\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f_{2}(s_{n})\geq \limsup _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f(s_{n})}
1280:{\displaystyle {\frac {1}{b-a}}\int _{a}^{b}f_{1}(x)\,dx=\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f_{1}(s_{n})\leq \liminf _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f(s_{n})}
3663:
3218:
2989:
2839:
2905:
2448:
123:
4528:
Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study
Institute on equidistribution in number theory, Montréal, Canada, July 11–22, 2005
3416:
646:
3857:
3890:
3436:
3332:
3268:
2238:
3016:
3937:
3910:
3163:
3816:. Clearly every well-distributed sequence is uniformly distributed, but the converse does not hold. The definition of well-distributed modulo 1 is analogous.
3242:
1588:
Finally, for complex-valued
Riemann-integrable functions, the result follows again from linearity, and from the fact that every such function can be written as
2937:
1957:
954:
1649:
of some equidistributed sequence. Then in the criterion, the left hand side is always 1, whereas the right hand side is zero, because the sequence is
1585:
differ by at most ε. Since ε is arbitrary, we have the existence of the limit, and by
Darboux's definition of the integral, it is the correct limit.
522:{\displaystyle D_{N}=\sup _{a\leq c\leq d\leq b}\left\vert {\frac {\left|\{\,s_{1},\dots ,s_{N}\,\}\cap \right|}{N}}-{\frac {d-c}{b-a}}\right\vert .}
3945:
1627:
It is not possible to generalize the integral criterion to a class of functions bigger than just the
Riemann-integrable ones. For example, if the
4036:
space, there exists an equidistributed sequence with respect to the measure; indeed, this follows immediately from the fact that such a space is
4579:
4539:
4306:
68:, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in
4666:
4291:
Twentieth century harmonic analysis–a celebration. Proceedings of the NATO Advanced Study
Institute, Il Ciocco, Italy, July 2–15, 2000
1519:
4501:
4467:
4444:
1513:
828:, then the left hand side is the proportion of points of the sequence falling in the interval , and the right hand side is exactly
4070:
2201:
2768:
then this sequence is not equidistributed modulo 1, because there are only a finite number of options for the fractional part of
2077:
If the sequence is equidistributed modulo 1, then we can apply the
Riemann integral criterion (described above) on the function
4567:
4485:
4331:
4279:
3021:
3337:
1624:, where integrals are computed by sampling the function over a sequence of random variables equidistributed in the interval.
2151:
4571:
4493:
788:{\displaystyle \lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f\left(s_{n}\right)={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx}
2080:
291:
For example, if a sequence is equidistributed in , since the interval occupies 1/5 of the length of the interval , as
831:
4671:
4037:
3825:
69:
4065:
4060:
3276:
2352:
2147:
1787:
810:
First note that the definition of an equidistributed sequence is equivalent to the integral criterion whenever
550:
4293:. NATO Sci. Ser. II, Math. Phys. Chem. Vol. 33. Dordrecht: Kluwer Academic Publishers. pp. 271–293.
3799:{\displaystyle \lim _{n\to \infty }{\left|\{\,s_{k+1},\dots ,s_{k+n}\,\}\cap \right| \over n}={d-c \over b-a}}
4676:
4375:
4198:
3916:
3447:
1923:
1621:
618:
73:
4651:
250:{\displaystyle \lim _{n\to \infty }{\left|\{\,s_{1},\dots ,s_{n}\,\}\cap \right| \over n}={d-c \over b-a}.}
4384:
4336:"Über eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem"
3171:
2942:
2792:
1910:
566:
114:
4340:
4156:
2414:
4561:
4048:
3913:
2848:
2138:
Conversely, suppose Weyl's criterion holds. Then the
Riemann integral criterion holds for functions
4455:
4432:
4025:
3395:
629:
4357:
4180:
1646:
893:
815:
4043:
The general phenomenon of equidistribution comes up a lot for dynamical systems associated with
2336:{\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=0}^{n-1}e^{2\pi i\ell \cdot v_{j}}=0.}
610:, ...) is a sequence contained in the interval . Then the following conditions are equivalent:
4624:
4605:
4575:
4535:
4497:
4463:
4440:
4302:
4222:
3830:
1791:
1761:
1658:
1628:
582:
4608:
3862:
3421:
3317:
3247:
1672:
is a function such that the criterion above holds for any equidistributed sequence in , then
4585:
4545:
4507:
4473:
4401:
4393:
4349:
4312:
4294:
4283:
4238:
4230:
4212:
4203:
4172:
3553:
2451:
1914:
1863:
912:
570:
2994:
4589:
4549:
4531:
4511:
4477:
4405:
4397:
4316:
4242:
4234:
4029:
3922:
3895:
3129:
2765:
2061:
1714:
310:
4530:. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 237. Dordrecht:
2046:{\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}e^{2\pi i\ell a_{j}}=0.}
1838:
with at least one coefficient other than the constant term irrational then the sequence
3227:
2757:
tend to infinity, the left hand side tends to zero, and Weyl's criterion is satisfied.
3314:
is uniformly distributed mod 1 (or 1-uniformly distributed) for any irrational number
2910:
1850:
This was proven by Weyl and is an application of van der Corput's difference theorem.
1056:{\displaystyle \textstyle \int _{a}^{b}(f_{2}(x)-f_{1}(x))\,dx\leq \varepsilon (b-a).}
4660:
4361:
4184:
4022:
2211:, assuming the natural generalization of the definition of equidistribution modulo 1:
1650:
897:
17:
4627:
4152:
2136:
which has integral zero on the interval . This gives Weyl's criterion immediately.
2057:
1874:
877:
This means 2 ⇒ 1 (since indicator functions are Riemann-integrable), and 1 ⇒ 2 for
4490:
Ten lectures on the interface between analytic number theory and harmonic analysis
4379:
4298:
2228:
is equidistributed modulo 1 if and only if for any non-zero vector ℓ ∈
4557:
574:
106:
57:
28:
4645:
4033:
4011:{\displaystyle {\frac {\sum _{k=1}^{n}\delta _{x_{k}}}{n}}\Rightarrow \mu \ .}
3541:
1835:
4226:
4632:
4613:
4201:(1931), "Diophantische Ungleichungen. I. Zur Gleichverteilung Modulo Eins",
4044:
3621:
2208:
4492:. Regional Conference Series in Mathematics. Vol. 84. Providence, RI:
3608:) are equidistributed mod 1. However it is known that the sequence (
1636:
32:
4641:
4098:
3334:, but is never even 2-uniformly distributed. In contrast, the sequence
2182:, thereby proving equidistribution modulo 1 of the given sequence.
4353:
4217:
4176:
1948:
3540:
Metric theorems describe the behaviour of a parametrised sequence for
2183:
1609:
4380:"Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins"
4335:
2169:
be the indicator function of an interval. It is possible to bound
2060:. It allows equidistribution questions to be reduced to bounds on
1578:{\displaystyle \textstyle {\frac {1}{N}}\sum _{n=1}^{N}f(s_{n})}
4652:
Lecture notes by Charles Walkden with proof of Weyl's Criterion
2450:
can never be 1. Using the formula for the sum of a finite
3820:
Sequences equidistributed with respect to an arbitrary measure
1947:
is equidistributed modulo 1 if and only if for all non-zero
884:
Note that both sides of the integral criterion equation are
2056:
The criterion is named after, and was first formulated by,
2142:
as above, and by linearity of the criterion, it holds for
1909:
is equidistributed modulo 1. This is a famous theorem of
907:
being a general Riemann-integrable function, first assume
3119:{\displaystyle b_{n}:=(a'_{n+1},\dots ,a'_{n+k})\in ^{k}}
3385:{\displaystyle (\alpha ,\alpha ^{2},\alpha ^{3},\dots )}
2200:
A quantitative form of Weyl's criterion is given by the
1862:
uniformly distributed modulo 1. This fact is related to
313:; rather, it is a determinate sequence of real numbers.
1641:, then this criterion fails. As a counterexample, take
276:} ∩ | denotes the number of elements, out of the first
4284:"Harmonic analysis as found in analytic number theory"
2417:
2084:
2068:
1523:
958:
835:
801:
532:
A sequence is thus equidistributed if the discrepancy
3948:
3925:
3898:
3865:
3833:
3666:
3424:
3398:
3340:
3320:
3279:
3250:
3230:
3174:
3132:
3024:
2997:
2945:
2913:
2851:
2795:
2463:
2241:
2083:
1960:
1522:
1294:
1072:
957:
834:
661:
632:
363:
126:
4159:[On the distribution of numbers modulo one]
3592:) is equidistributed mod 1 for almost all values of
3581:) is equidistributed mod 1 for almost all values of
573:
in the interval , then its integral is the limit of
3392:is completely uniformly distributed for almost all
2399:starts from 0, to simplify the formula later). Let
2154:and an approximation argument, this extends to any
4010:
3931:
3904:
3884:
3851:
3798:
3430:
3410:
3384:
3326:
3306:
3262:
3236:
3212:
3157:
3118:
3010:
2983:
2931:
2899:
2833:
2734:
2442:
2335:
2128:
2045:
1765:Illustration of the filling of the unit interval (
1577:
1501:
1279:
1055:
868:
787:
640:
521:
249:
2351:Weyl's criterion can be used to easily prove the
2129:{\displaystyle \textstyle f(x)=e^{2\pi i\ell x},}
4157:"Über die Gleichverteilung von Zahlen mod. Eins"
3668:
2243:
1962:
1434:
1358:
1212:
1136:
663:
378:
128:
4526:Granville, Andrew; Rudnick, Zeév, eds. (2007).
3526:is uniformly distributed modulo 1, then so is s
3244:-uniformly distributed for each natural number
1869:The sequence of all multiples of an irrational
1781:-axis). Gradation in colour is due to aliasing.
1668:states the converse of the above criterion: If
557:Riemann integral criterion for equidistribution
3892:is said to be equidistributed with respect to
3476:is uniformly distributed modulo 1, then so is
869:{\displaystyle \textstyle {\frac {d-c}{b-a}}.}
4417:Kuipers & Niederreiter (2006) p. 171
4142:Kuipers & Niederreiter (2006) p. 127
4133:Kuipers & Niederreiter (2006) p. 129
2845:if not only the sequence of fractional parts
2207:Weyl's criterion extends naturally to higher
1608:are real-valued and Riemann-integrable.
8:
4254:Kuipers & Niederreiter (2006) p. 26
4124:Kuipers & Niederreiter (2006) p. 27
3736:
3690:
2355:, stating that the sequence of multiples 0,
451:
417:
184:
150:
4115:Kuipers & Niederreiter (2006) p. 8
4047:, for example in Margulis' solution to the
2371:is equidistributed modulo 1 if and only if
295:becomes large, the proportion of the first
280:elements of the sequence, that are between
4274:
4272:
1773:terms of the Van der Corput sequence, for
4216:
4089:Kuipers & Niederreiter (2006) pp. 2–3
3982:
3977:
3967:
3956:
3949:
3947:
3924:
3897:
3873:
3864:
3832:
3770:
3735:
3723:
3698:
3693:
3683:
3671:
3665:
3423:
3397:
3367:
3354:
3339:
3319:
3307:{\displaystyle (\alpha ,2\alpha ,\dots )}
3278:
3249:
3229:
3222:completely uniformly distributed mod 1
3195:
3182:
3173:
3149:
3131:
3110:
3073:
3045:
3029:
3023:
3002:
2996:
2966:
2953:
2944:
2912:
2888:
2872:
2856:
2850:
2816:
2803:
2794:
2704:
2684:
2656:
2623:
2610:
2592:
2570:
2549:
2538:
2500:
2484:
2473:
2462:
2422:
2416:
2382:is irrational and denote our sequence by
2319:
2299:
2283:
2272:
2258:
2246:
2240:
2104:
2082:
2029:
2012:
2002:
1991:
1977:
1965:
1959:
1565:
1549:
1538:
1524:
1521:
1490:
1474:
1463:
1449:
1437:
1421:
1408:
1398:
1387:
1373:
1361:
1347:
1332:
1322:
1317:
1295:
1293:
1268:
1252:
1241:
1227:
1215:
1199:
1186:
1176:
1165:
1151:
1139:
1125:
1110:
1100:
1095:
1073:
1071:
1021:
1003:
981:
968:
963:
956:
836:
833:
778:
760:
755:
733:
720:
703:
692:
678:
666:
660:
634:
633:
631:
485:
450:
444:
425:
420:
410:
381:
368:
362:
218:
183:
177:
158:
153:
143:
131:
125:
4262:
4260:
3657:on if for any subinterval of we have
1760:
1753:⌋, is equidistributed in the interval .
892:, and therefore the criterion holds for
354:, ...) with respect to the interval as
4082:
3600:It is not known whether the sequences (
2745:a finite bound that does not depend on
3560:For any sequence of distinct integers
1790:: The sequence of all multiples of an
117:if for every subinterval of we have
4111:
4109:
4107:
4099:http://math.uga.edu/~pete/udnotes.pdf
3653:, ...) of real numbers is said to be
3552:not lying in some exceptional set of
1705:, ...) of real numbers is said to be
919: > 0 two step functions
7:
3213:{\displaystyle (a_{1},a_{2},\dots )}
2984:{\displaystyle (b_{1},b_{2},\dots )}
2834:{\displaystyle (a_{1},a_{2},\dots )}
2403: ≠ 0 be an integer. Since
2064:, a fundamental and general method.
1846:) is uniformly distributed modulo 1.
1620:This criterion leads to the idea of
614:The sequence is equidistributed on .
3442:van der Corput's difference theorem
2443:{\textstyle e^{2\pi i\ell \alpha }}
915:of the integral, we have for every
3678:
3496:of integers such that if for each
3441:
2253:
1972:
1444:
1368:
1222:
1146:
673:
138:
25:
4570:. Vol. 142. Providence, RI:
4460:Uniform Distribution of Sequences
4437:Uniform Distribution of Sequences
4211:, Springer Netherlands: 373–456,
3438:except for a set of measure 0).
1514:limit superior and limit inferior
896:of interval indicators, that is,
2843:k-uniformly distributed mod 1
1512:By subtracting, we see that the
72:theory and have applications to
4568:Graduate Studies in Mathematics
2749:. Therefore, after dividing by
577:taken by sampling the function
3996:
3879:
3866:
3846:
3834:
3754:
3742:
3675:
3616:equidistributed mod 1 if
3379:
3341:
3301:
3280:
3220:of real numbers is said to be
3207:
3175:
3146:
3133:
3126:, is uniformly distributed in
3107:
3094:
3088:
3038:
2978:
2946:
2926:
2914:
2900:{\displaystyle a_{n}':=a_{n}-}
2894:
2881:
2841:of real numbers is said to be
2828:
2796:
2250:
2094:
2088:
1969:
1711:uniformly distributed modulo 1
1571:
1558:
1496:
1483:
1441:
1427:
1414:
1365:
1344:
1338:
1274:
1261:
1219:
1205:
1192:
1143:
1122:
1116:
1046:
1034:
1018:
1015:
1009:
993:
987:
974:
911:is real-valued. Then by using
775:
769:
670:
617:For every Riemann-integrable (
469:
457:
202:
190:
135:
1:
4572:American Mathematical Society
4563:Higher order Fourier analysis
4494:American Mathematical Society
4462:. John Wiley & Sons Inc.
4289:. In Byrnes, James S. (ed.).
2785:Complete uniform distribution
4299:10.1007/978-94-010-0662-0_13
4266:Montgomery (1994) p. 18
3411:{\displaystyle \alpha >1}
2907:is uniformly distributed in
2411:can never be an integer, so
1825:is equidistributed modulo 1.
649:, the following limit holds:
641:{\displaystyle \mathbb {C} }
2760:Conversely, notice that if
1676:is Riemann-integrable in .
4693:
4609:"Equidistributed Sequence"
3273:For example, the sequence
2367:, ... of some real number
4667:Diophantine approximation
3826:probability measure space
3628:Well-distributed sequence
3548:: that is, for values of
3544:values of some parameter
2152:Stone–Weierstrass theorem
1938:states that the sequence
1930:
1680:Equidistribution modulo 1
70:Diophantine approximation
4066:Low-discrepancy sequence
4061:Equidistribution theorem
3852:{\displaystyle (X,\mu )}
3450:states that if for each
2353:equidistribution theorem
2148:trigonometric polynomial
1788:equidistribution theorem
1707:equidistributed modulo 1
551:low-discrepancy sequence
3885:{\displaystyle (x_{n})}
3859:, a sequence of points
3448:Johannes van der Corput
3431:{\displaystyle \alpha }
3327:{\displaystyle \alpha }
3263:{\displaystyle k\geq 1}
1924:van der Corput sequence
1769:-axis) using the first
1713:if the sequence of the
1622:Monte-Carlo integration
74:Monte Carlo integration
4439:. Dover Publications.
4385:Compositio Mathematica
4071:Erdős–Turán inequality
4012:
3972:
3933:
3906:
3886:
3853:
3800:
3432:
3412:
3386:
3328:
3308:
3264:
3238:
3214:
3159:
3120:
3012:
2985:
2939:but also the sequence
2933:
2901:
2835:
2736:
2560:
2495:
2444:
2337:
2294:
2202:Erdős–Turán inequality
2130:
2047:
2007:
1911:analytic number theory
1782:
1666:de Bruijn–Post Theorem
1579:
1554:
1503:
1479:
1403:
1281:
1257:
1181:
1057:
870:
789:
708:
642:
523:
260:(Here, the notation |{
251:
4341:Mathematische Annalen
4013:
3952:
3934:
3907:
3887:
3854:
3801:
3433:
3413:
3387:
3329:
3309:
3265:
3239:
3215:
3160:
3121:
3013:
3011:{\displaystyle b_{n}}
2986:
2934:
2902:
2836:
2737:
2534:
2469:
2445:
2338:
2268:
2131:
2048:
1987:
1764:
1580:
1534:
1504:
1459:
1383:
1282:
1237:
1161:
1058:
903:To show it holds for
871:
790:
688:
643:
524:
252:
66:uniformly distributed
18:Weyl's criterion
4049:Oppenheim conjecture
3946:
3932:{\displaystyle \mu }
3923:
3905:{\displaystyle \mu }
3896:
3863:
3831:
3664:
3422:
3396:
3338:
3318:
3277:
3248:
3228:
3172:
3158:{\displaystyle ^{k}}
3130:
3022:
2995:
2943:
2911:
2849:
2793:
2461:
2415:
2239:
2081:
1958:
1520:
1292:
1070:
955:
913:Darboux's definition
832:
659:
630:
361:
124:
113:on a non-degenerate
4486:Montgomery, Hugh L.
4280:Montgomery, Hugh L.
4026:probability measure
3087:
3059:
2864:
1926:is equidistributed.
1830:More generally, if
1327:
1105:
973:
894:linear combinations
818:of an interval: If
765:
545:tends to infinity.
309:) is a sequence of
4628:"Weyl's Criterion"
4625:Weisstein, Eric W.
4606:Weisstein, Eric W.
4354:10.1007/BF01456856
4218:10.1007/BF02545780
4199:van der Corput, J.
4177:10.1007/BF01475864
4155:(September 1916).
4008:
3929:
3902:
3882:
3849:
3796:
3682:
3490:van der Corput set
3428:
3408:
3382:
3324:
3304:
3260:
3234:
3210:
3155:
3116:
3069:
3041:
3008:
2981:
2929:
2897:
2852:
2831:
2732:
2440:
2333:
2257:
2176:interval indicator
2126:
2125:
2043:
1976:
1783:
1647:indicator function
1635:is taken to be in
1631:is considered and
1575:
1574:
1499:
1448:
1372:
1313:
1277:
1226:
1150:
1091:
1053:
1052:
959:
866:
865:
816:indicator function
785:
751:
677:
638:
519:
404:
247:
142:
4672:Dynamical systems
4581:978-0-8218-8986-2
4541:978-1-4020-5403-7
4308:978-0-7923-7169-4
4004:
3994:
3824:For an arbitrary
3794:
3765:
3667:
3237:{\displaystyle k}
2727:
2675:
2266:
2242:
2190:
2189:
1985:
1961:
1854:The sequence log(
1659:almost everywhere
1629:Lebesgue integral
1616:
1615:
1532:
1457:
1433:
1381:
1357:
1311:
1235:
1211:
1159:
1135:
1089:
860:
749:
686:
662:
541:tends to zero as
509:
480:
377:
242:
213:
127:
16:(Redirected from
4684:
4642:Weyl's Criterion
4638:
4637:
4619:
4618:
4593:
4553:
4515:
4481:
4456:Niederreiter, H.
4450:
4433:Niederreiter, H.
4418:
4415:
4409:
4408:
4372:
4366:
4364:
4332:Bernstein, Felix
4327:
4321:
4320:
4288:
4276:
4267:
4264:
4255:
4252:
4246:
4245:
4220:
4204:Acta Mathematica
4195:
4189:
4188:
4162:
4149:
4143:
4140:
4134:
4131:
4125:
4122:
4116:
4113:
4102:
4096:
4090:
4087:
4017:
4015:
4014:
4009:
4002:
3995:
3990:
3989:
3988:
3987:
3986:
3971:
3966:
3950:
3938:
3936:
3935:
3930:
3917:converges weakly
3911:
3909:
3908:
3903:
3891:
3889:
3888:
3883:
3878:
3877:
3858:
3856:
3855:
3850:
3805:
3803:
3802:
3797:
3795:
3793:
3782:
3771:
3766:
3761:
3757:
3734:
3733:
3709:
3708:
3684:
3681:
3655:well-distributed
3607:
3569:, the sequence (
3554:Lebesgue measure
3437:
3435:
3434:
3429:
3417:
3415:
3414:
3409:
3391:
3389:
3388:
3383:
3372:
3371:
3359:
3358:
3333:
3331:
3330:
3325:
3313:
3311:
3310:
3305:
3269:
3267:
3266:
3261:
3243:
3241:
3240:
3235:
3219:
3217:
3216:
3211:
3200:
3199:
3187:
3186:
3164:
3162:
3161:
3156:
3154:
3153:
3125:
3123:
3122:
3117:
3115:
3114:
3083:
3055:
3034:
3033:
3017:
3015:
3014:
3009:
3007:
3006:
2990:
2988:
2987:
2982:
2971:
2970:
2958:
2957:
2938:
2936:
2935:
2932:{\displaystyle }
2930:
2906:
2904:
2903:
2898:
2893:
2892:
2877:
2876:
2860:
2840:
2838:
2837:
2832:
2821:
2820:
2808:
2807:
2741:
2739:
2738:
2733:
2728:
2726:
2722:
2721:
2720:
2685:
2680:
2676:
2674:
2673:
2672:
2644:
2643:
2642:
2611:
2602:
2598:
2597:
2596:
2591:
2587:
2586:
2559:
2548:
2525:
2521:
2520:
2519:
2494:
2483:
2452:geometric series
2449:
2447:
2446:
2441:
2439:
2438:
2347:Example of usage
2342:
2340:
2339:
2334:
2326:
2325:
2324:
2323:
2293:
2282:
2267:
2259:
2256:
2135:
2133:
2132:
2127:
2121:
2120:
2072:Sketch of proof
2069:
2062:exponential sums
2052:
2050:
2049:
2044:
2036:
2035:
2034:
2033:
2006:
2001:
1986:
1978:
1975:
1936:Weyl's criterion
1931:Weyl's criterion
1915:I. M. Vinogradov
1715:fractional parts
1584:
1582:
1581:
1576:
1570:
1569:
1553:
1548:
1533:
1525:
1508:
1506:
1505:
1500:
1495:
1494:
1478:
1473:
1458:
1450:
1447:
1426:
1425:
1413:
1412:
1402:
1397:
1382:
1374:
1371:
1337:
1336:
1326:
1321:
1312:
1310:
1296:
1286:
1284:
1283:
1278:
1273:
1272:
1256:
1251:
1236:
1228:
1225:
1204:
1203:
1191:
1190:
1180:
1175:
1160:
1152:
1149:
1115:
1114:
1104:
1099:
1090:
1088:
1074:
1062:
1060:
1059:
1054:
1008:
1007:
986:
985:
972:
967:
875:
873:
872:
867:
861:
859:
848:
837:
802:
794:
792:
791:
786:
764:
759:
750:
748:
734:
729:
725:
724:
707:
702:
687:
679:
676:
648:
647:
645:
644:
639:
637:
571:Riemann integral
528:
526:
525:
520:
515:
511:
510:
508:
497:
486:
481:
476:
472:
449:
448:
430:
429:
411:
403:
373:
372:
333:for a sequence (
311:random variables
256:
254:
253:
248:
243:
241:
230:
219:
214:
209:
205:
182:
181:
163:
162:
144:
141:
21:
4692:
4691:
4687:
4686:
4685:
4683:
4682:
4681:
4657:
4656:
4623:
4622:
4604:
4603:
4600:
4582:
4556:
4542:
4532:Springer-Verlag
4525:
4522:
4520:Further reading
4504:
4484:
4470:
4453:
4447:
4430:
4427:
4422:
4421:
4416:
4412:
4374:
4373:
4369:
4330:
4328:
4324:
4309:
4286:
4278:
4277:
4270:
4265:
4258:
4253:
4249:
4197:
4196:
4192:
4160:
4151:
4150:
4146:
4141:
4137:
4132:
4128:
4123:
4119:
4114:
4105:
4097:
4093:
4088:
4084:
4079:
4057:
3978:
3973:
3951:
3944:
3943:
3921:
3920:
3912:if the mean of
3894:
3893:
3869:
3861:
3860:
3829:
3828:
3822:
3783:
3772:
3719:
3694:
3689:
3685:
3662:
3661:
3652:
3645:
3638:
3630:
3605:
3577:
3568:
3538:
3536:Metric theorems
3531:
3525:
3516:
3484:
3475:
3466:
3444:
3420:
3419:
3418:(i.e., for all
3394:
3393:
3363:
3350:
3336:
3335:
3316:
3315:
3275:
3274:
3246:
3245:
3226:
3225:
3191:
3178:
3170:
3169:
3145:
3128:
3127:
3106:
3025:
3020:
3019:
2998:
2993:
2992:
2962:
2949:
2941:
2940:
2909:
2908:
2884:
2868:
2847:
2846:
2812:
2799:
2791:
2790:
2787:
2776:
2700:
2693:
2689:
2652:
2645:
2619:
2612:
2606:
2566:
2562:
2561:
2533:
2529:
2496:
2468:
2464:
2459:
2458:
2418:
2413:
2412:
2407:is irrational,
2390:
2375:is irrational.
2349:
2315:
2295:
2237:
2236:
2223:
2197:
2195:Generalizations
2100:
2079:
2078:
2025:
2008:
1956:
1955:
1946:
1933:
1913:, published by
1777:from 0 to 999 (
1759:
1752:
1743:
1734:
1725:
1704:
1697:
1690:
1682:
1561:
1518:
1517:
1486:
1417:
1404:
1328:
1300:
1290:
1289:
1264:
1195:
1182:
1106:
1078:
1068:
1067:
999:
977:
953:
952:
950:
939:
932:
925:
849:
838:
830:
829:
827:
738:
716:
712:
657:
656:
628:
627:
622:
609:
602:
595:
561:Recall that if
559:
540:
498:
487:
440:
421:
416:
412:
409:
405:
364:
359:
358:
353:
346:
339:
332:
319:
308:
275:
266:
231:
220:
173:
154:
149:
145:
122:
121:
111:equidistributed
104:
97:
90:
82:
62:equidistributed
55:
48:
41:
23:
22:
15:
12:
11:
5:
4690:
4688:
4680:
4679:
4677:Ergodic theory
4674:
4669:
4659:
4658:
4655:
4654:
4649:
4639:
4620:
4599:
4598:External links
4596:
4595:
4594:
4580:
4554:
4540:
4521:
4518:
4517:
4516:
4502:
4482:
4468:
4451:
4445:
4426:
4423:
4420:
4419:
4410:
4367:
4348:(3): 417–439,
4322:
4307:
4268:
4256:
4247:
4190:
4171:(3): 313–352.
4144:
4135:
4126:
4117:
4103:
4091:
4081:
4080:
4078:
4075:
4074:
4073:
4068:
4063:
4056:
4053:
4019:
4018:
4007:
4001:
3998:
3993:
3985:
3981:
3976:
3970:
3965:
3962:
3959:
3955:
3928:
3914:point measures
3901:
3881:
3876:
3872:
3868:
3848:
3845:
3842:
3839:
3836:
3821:
3818:
3807:
3806:
3792:
3789:
3786:
3781:
3778:
3775:
3769:
3764:
3760:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3732:
3729:
3726:
3722:
3718:
3715:
3712:
3707:
3704:
3701:
3697:
3692:
3688:
3680:
3677:
3674:
3670:
3650:
3643:
3636:
3629:
3626:
3598:
3597:
3588:The sequence (
3586:
3573:
3564:
3537:
3534:
3527:
3521:
3508:
3480:
3471:
3458:
3443:
3440:
3427:
3407:
3404:
3401:
3381:
3378:
3375:
3370:
3366:
3362:
3357:
3353:
3349:
3346:
3343:
3323:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3259:
3256:
3253:
3233:
3209:
3206:
3203:
3198:
3194:
3190:
3185:
3181:
3177:
3152:
3148:
3144:
3141:
3138:
3135:
3113:
3109:
3105:
3102:
3099:
3096:
3093:
3090:
3086:
3082:
3079:
3076:
3072:
3068:
3065:
3062:
3058:
3054:
3051:
3048:
3044:
3040:
3037:
3032:
3028:
3018:is defined as
3005:
3001:
2980:
2977:
2974:
2969:
2965:
2961:
2956:
2952:
2948:
2928:
2925:
2922:
2919:
2916:
2896:
2891:
2887:
2883:
2880:
2875:
2871:
2867:
2863:
2859:
2855:
2830:
2827:
2824:
2819:
2815:
2811:
2806:
2802:
2798:
2786:
2783:
2772:
2743:
2742:
2731:
2725:
2719:
2716:
2713:
2710:
2707:
2703:
2699:
2696:
2692:
2688:
2683:
2679:
2671:
2668:
2665:
2662:
2659:
2655:
2651:
2648:
2641:
2638:
2635:
2632:
2629:
2626:
2622:
2618:
2615:
2609:
2605:
2601:
2595:
2590:
2585:
2582:
2579:
2576:
2573:
2569:
2565:
2558:
2555:
2552:
2547:
2544:
2541:
2537:
2532:
2528:
2524:
2518:
2515:
2512:
2509:
2506:
2503:
2499:
2493:
2490:
2487:
2482:
2479:
2476:
2472:
2467:
2437:
2434:
2431:
2428:
2425:
2421:
2386:
2348:
2345:
2344:
2343:
2332:
2329:
2322:
2318:
2314:
2311:
2308:
2305:
2302:
2298:
2292:
2289:
2286:
2281:
2278:
2275:
2271:
2265:
2262:
2255:
2252:
2249:
2245:
2224:of vectors in
2219:
2213:
2212:
2205:
2196:
2193:
2192:
2191:
2188:
2187:
2124:
2119:
2116:
2113:
2110:
2107:
2103:
2099:
2096:
2093:
2090:
2087:
2074:
2073:
2054:
2053:
2042:
2039:
2032:
2028:
2024:
2021:
2018:
2015:
2011:
2005:
2000:
1997:
1994:
1990:
1984:
1981:
1974:
1971:
1968:
1964:
1942:
1932:
1929:
1928:
1927:
1919:
1918:
1907:
1906:
1905:
1879:
1878:
1873:by successive
1867:
1848:
1847:
1827:
1826:
1823:
1822:
1821:
1799:
1798:
1758:
1755:
1748:
1744: − ⌊
1739:
1730:
1726:, denoted by (
1721:
1702:
1695:
1688:
1681:
1678:
1618:
1617:
1614:
1613:
1573:
1568:
1564:
1560:
1557:
1552:
1547:
1544:
1541:
1537:
1531:
1528:
1510:
1509:
1498:
1493:
1489:
1485:
1482:
1477:
1472:
1469:
1466:
1462:
1456:
1453:
1446:
1443:
1440:
1436:
1435:lim sup
1432:
1429:
1424:
1420:
1416:
1411:
1407:
1401:
1396:
1393:
1390:
1386:
1380:
1377:
1370:
1367:
1364:
1360:
1356:
1353:
1350:
1346:
1343:
1340:
1335:
1331:
1325:
1320:
1316:
1309:
1306:
1303:
1299:
1287:
1276:
1271:
1267:
1263:
1260:
1255:
1250:
1247:
1244:
1240:
1234:
1231:
1224:
1221:
1218:
1214:
1213:lim inf
1210:
1207:
1202:
1198:
1194:
1189:
1185:
1179:
1174:
1171:
1168:
1164:
1158:
1155:
1148:
1145:
1142:
1138:
1134:
1131:
1128:
1124:
1121:
1118:
1113:
1109:
1103:
1098:
1094:
1087:
1084:
1081:
1077:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1020:
1017:
1014:
1011:
1006:
1002:
998:
995:
992:
989:
984:
980:
976:
971:
966:
962:
948:
937:
930:
923:
898:step functions
864:
858:
855:
852:
847:
844:
841:
826:
807:
806:
798:
797:
796:
795:
784:
781:
777:
774:
771:
768:
763:
758:
754:
747:
744:
741:
737:
732:
728:
723:
719:
715:
711:
706:
701:
698:
695:
691:
685:
682:
675:
672:
669:
665:
651:
650:
636:
619:complex-valued
615:
607:
600:
593:
558:
555:
536:
530:
529:
518:
514:
507:
504:
501:
496:
493:
490:
484:
479:
475:
471:
468:
465:
462:
459:
456:
453:
447:
443:
439:
436:
433:
428:
424:
419:
415:
408:
402:
399:
396:
393:
390:
387:
384:
380:
376:
371:
367:
351:
344:
337:
328:
321:We define the
318:
315:
304:
271:
264:
258:
257:
246:
240:
237:
234:
229:
226:
223:
217:
212:
208:
204:
201:
198:
195:
192:
189:
186:
180:
176:
172:
169:
166:
161:
157:
152:
148:
140:
137:
134:
130:
109:is said to be
102:
95:
88:
81:
78:
60:is said to be
53:
46:
39:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4689:
4678:
4675:
4673:
4670:
4668:
4665:
4664:
4662:
4653:
4650:
4647:
4643:
4640:
4635:
4634:
4629:
4626:
4621:
4616:
4615:
4610:
4607:
4602:
4601:
4597:
4591:
4587:
4583:
4577:
4573:
4569:
4565:
4564:
4559:
4555:
4551:
4547:
4543:
4537:
4533:
4529:
4524:
4523:
4519:
4513:
4509:
4505:
4503:0-8218-0737-4
4499:
4495:
4491:
4487:
4483:
4479:
4475:
4471:
4469:0-471-51045-9
4465:
4461:
4457:
4454:Kuipers, L.;
4452:
4448:
4446:0-486-45019-8
4442:
4438:
4434:
4431:Kuipers, L.;
4429:
4428:
4424:
4414:
4411:
4407:
4403:
4399:
4395:
4391:
4387:
4386:
4381:
4377:
4376:Koksma, J. F.
4371:
4368:
4363:
4359:
4355:
4351:
4347:
4343:
4342:
4337:
4333:
4326:
4323:
4318:
4314:
4310:
4304:
4300:
4296:
4292:
4285:
4281:
4275:
4273:
4269:
4263:
4261:
4257:
4251:
4248:
4244:
4240:
4236:
4232:
4228:
4224:
4219:
4214:
4210:
4206:
4205:
4200:
4194:
4191:
4186:
4182:
4178:
4174:
4170:
4167:(in German).
4166:
4158:
4154:
4148:
4145:
4139:
4136:
4130:
4127:
4121:
4118:
4112:
4110:
4108:
4104:
4100:
4095:
4092:
4086:
4083:
4076:
4072:
4069:
4067:
4064:
4062:
4059:
4058:
4054:
4052:
4050:
4046:
4041:
4039:
4035:
4031:
4027:
4024:
4005:
3999:
3991:
3983:
3979:
3974:
3968:
3963:
3960:
3957:
3953:
3942:
3941:
3940:
3926:
3918:
3915:
3899:
3874:
3870:
3843:
3840:
3837:
3827:
3819:
3817:
3815:
3811:
3790:
3787:
3784:
3779:
3776:
3773:
3767:
3762:
3758:
3751:
3748:
3745:
3739:
3730:
3727:
3724:
3720:
3716:
3713:
3710:
3705:
3702:
3699:
3695:
3686:
3672:
3660:
3659:
3658:
3656:
3649:
3642:
3635:
3627:
3625:
3623:
3619:
3615:
3611:
3603:
3595:
3591:
3587:
3584:
3580:
3576:
3572:
3567:
3563:
3559:
3558:
3557:
3555:
3551:
3547:
3543:
3535:
3533:
3530:
3524:
3520:
3515:
3511:
3507:
3504:the sequence
3503:
3499:
3495:
3491:
3486:
3483:
3479:
3474:
3470:
3465:
3461:
3457:
3454:the sequence
3453:
3449:
3446:A theorem of
3439:
3425:
3405:
3402:
3399:
3376:
3373:
3368:
3364:
3360:
3355:
3351:
3347:
3344:
3321:
3298:
3295:
3292:
3289:
3286:
3283:
3271:
3257:
3254:
3251:
3231:
3223:
3204:
3201:
3196:
3192:
3188:
3183:
3179:
3166:
3150:
3142:
3139:
3136:
3111:
3103:
3100:
3097:
3091:
3084:
3080:
3077:
3074:
3070:
3066:
3063:
3060:
3056:
3052:
3049:
3046:
3042:
3035:
3030:
3026:
3003:
2999:
2975:
2972:
2967:
2963:
2959:
2954:
2950:
2923:
2920:
2917:
2889:
2885:
2878:
2873:
2869:
2865:
2861:
2857:
2853:
2844:
2825:
2822:
2817:
2813:
2809:
2804:
2800:
2784:
2782:
2780:
2777: =
2775:
2771:
2767:
2763:
2758:
2756:
2752:
2748:
2729:
2723:
2717:
2714:
2711:
2708:
2705:
2701:
2697:
2694:
2690:
2686:
2681:
2677:
2669:
2666:
2663:
2660:
2657:
2653:
2649:
2646:
2639:
2636:
2633:
2630:
2627:
2624:
2620:
2616:
2613:
2607:
2603:
2599:
2593:
2588:
2583:
2580:
2577:
2574:
2571:
2567:
2563:
2556:
2553:
2550:
2545:
2542:
2539:
2535:
2530:
2526:
2522:
2516:
2513:
2510:
2507:
2504:
2501:
2497:
2491:
2488:
2485:
2480:
2477:
2474:
2470:
2465:
2457:
2456:
2455:
2453:
2435:
2432:
2429:
2426:
2423:
2419:
2410:
2406:
2402:
2398:
2394:
2391: =
2389:
2385:
2381:
2376:
2374:
2370:
2366:
2362:
2358:
2354:
2346:
2330:
2327:
2320:
2316:
2312:
2309:
2306:
2303:
2300:
2296:
2290:
2287:
2284:
2279:
2276:
2273:
2269:
2263:
2260:
2247:
2235:
2234:
2233:
2231:
2227:
2222:
2218:
2215:The sequence
2210:
2206:
2203:
2199:
2198:
2194:
2186:
2185:
2181:
2177:
2172:
2168:
2165:Finally, let
2163:
2161:
2157:
2153:
2149:
2145:
2141:
2122:
2117:
2114:
2111:
2108:
2105:
2101:
2097:
2091:
2085:
2076:
2075:
2071:
2070:
2067:
2066:
2065:
2063:
2059:
2040:
2037:
2030:
2026:
2022:
2019:
2016:
2013:
2009:
2003:
1998:
1995:
1992:
1988:
1982:
1979:
1966:
1954:
1953:
1952:
1950:
1945:
1941:
1937:
1925:
1921:
1920:
1916:
1912:
1908:
1903:
1899:
1895:
1891:
1887:
1883:
1882:
1881:
1880:
1876:
1875:prime numbers
1872:
1868:
1865:
1864:Benford's law
1861:
1857:
1853:
1852:
1851:
1845:
1841:
1837:
1833:
1829:
1828:
1824:
1819:
1815:
1811:
1807:
1803:
1802:
1801:
1800:
1796:
1793:
1789:
1785:
1784:
1780:
1776:
1772:
1768:
1763:
1756:
1754:
1751:
1747:
1742:
1738:
1733:
1729:
1724:
1720:
1716:
1712:
1708:
1701:
1694:
1687:
1679:
1677:
1675:
1671:
1667:
1664:In fact, the
1662:
1660:
1656:
1652:
1648:
1644:
1640:
1639:
1634:
1630:
1625:
1623:
1612:
1611:
1607:
1603:
1599:
1595:
1592: =
1591:
1586:
1566:
1562:
1555:
1550:
1545:
1542:
1539:
1535:
1529:
1526:
1515:
1491:
1487:
1480:
1475:
1470:
1467:
1464:
1460:
1454:
1451:
1438:
1430:
1422:
1418:
1409:
1405:
1399:
1394:
1391:
1388:
1384:
1378:
1375:
1362:
1354:
1351:
1348:
1341:
1333:
1329:
1323:
1318:
1314:
1307:
1304:
1301:
1297:
1288:
1269:
1265:
1258:
1253:
1248:
1245:
1242:
1238:
1232:
1229:
1216:
1208:
1200:
1196:
1187:
1183:
1177:
1172:
1169:
1166:
1162:
1156:
1153:
1140:
1132:
1129:
1126:
1119:
1111:
1107:
1101:
1096:
1092:
1085:
1082:
1079:
1075:
1066:
1065:
1064:
1063:Notice that:
1049:
1043:
1040:
1037:
1031:
1028:
1025:
1022:
1012:
1004:
1000:
996:
990:
982:
978:
969:
964:
960:
947:
944: ≤
943:
940: ≤
936:
929:
922:
918:
914:
910:
906:
901:
899:
895:
891:
887:
882:
880:
862:
856:
853:
850:
845:
842:
839:
825:
821:
817:
813:
809:
808:
804:
803:
800:
799:
782:
779:
772:
766:
761:
756:
752:
745:
742:
739:
735:
730:
726:
721:
717:
713:
709:
704:
699:
696:
693:
689:
683:
680:
667:
655:
654:
653:
652:
625:
620:
616:
613:
612:
611:
606:
599:
592:
587:
584:
580:
576:
572:
568:
564:
556:
554:
552:
546:
544:
539:
535:
516:
512:
505:
502:
499:
494:
491:
488:
482:
477:
473:
466:
463:
460:
454:
445:
441:
437:
434:
431:
426:
422:
413:
406:
400:
397:
394:
391:
388:
385:
382:
374:
369:
365:
357:
356:
355:
350:
343:
336:
331:
327:
324:
316:
314:
312:
307:
303:
298:
294:
289:
287:
283:
279:
274:
270:
263:
244:
238:
235:
232:
227:
224:
221:
215:
210:
206:
199:
196:
193:
187:
178:
174:
170:
167:
164:
159:
155:
146:
132:
120:
119:
118:
116:
112:
108:
101:
94:
87:
79:
77:
75:
71:
67:
63:
59:
52:
45:
38:
34:
30:
19:
4631:
4612:
4562:
4558:Tao, Terence
4527:
4489:
4459:
4436:
4413:
4389:
4383:
4370:
4345:
4339:
4325:
4290:
4250:
4208:
4202:
4193:
4168:
4164:
4147:
4138:
4129:
4120:
4094:
4085:
4042:
4020:
3823:
3813:
3809:
3808:
3654:
3647:
3640:
3633:
3632:A sequence (
3631:
3617:
3613:
3609:
3601:
3599:
3593:
3589:
3582:
3578:
3574:
3570:
3565:
3561:
3549:
3545:
3539:
3528:
3522:
3518:
3513:
3509:
3505:
3501:
3497:
3493:
3489:
3487:
3481:
3477:
3472:
3468:
3463:
3459:
3455:
3451:
3445:
3272:
3221:
3167:
2842:
2788:
2778:
2773:
2769:
2761:
2759:
2754:
2753:and letting
2750:
2746:
2744:
2408:
2404:
2400:
2396:
2392:
2387:
2383:
2379:
2377:
2372:
2368:
2364:
2360:
2356:
2350:
2229:
2225:
2220:
2216:
2214:
2179:
2175:
2170:
2166:
2164:
2159:
2155:
2143:
2139:
2137:
2058:Hermann Weyl
2055:
1943:
1939:
1935:
1934:
1901:
1897:
1893:
1889:
1885:
1870:
1859:
1855:
1849:
1843:
1839:
1831:
1817:
1813:
1809:
1805:
1794:
1778:
1774:
1770:
1766:
1749:
1745:
1740:
1736:
1731:
1727:
1722:
1718:
1710:
1706:
1699:
1692:
1685:
1684:A sequence (
1683:
1673:
1669:
1665:
1663:
1654:
1642:
1637:
1632:
1626:
1619:
1605:
1601:
1597:
1593:
1589:
1587:
1511:
945:
941:
934:
927:
920:
916:
908:
904:
902:
889:
885:
883:
878:
876:
823:
819:
811:
623:
604:
597:
590:
588:
578:
575:Riemann sums
562:
560:
547:
542:
537:
533:
531:
348:
341:
334:
329:
325:
322:
320:
305:
301:
296:
292:
290:
285:
281:
277:
272:
268:
261:
259:
110:
107:real numbers
99:
92:
85:
84:A sequence (
83:
65:
61:
58:real numbers
50:
43:
36:
26:
4392:: 250–258,
4101:, Theorem 8
3168:A sequence
2789:A sequence
626: : →
621:) function
323:discrepancy
317:Discrepancy
29:mathematics
4661:Categories
4646:PlanetMath
4590:1277.11010
4550:1121.11004
4512:0814.11001
4478:0281.10001
4425:References
4406:0012.01401
4398:61.0205.01
4317:1001.11001
4243:0001.20102
4235:57.0230.05
4165:Math. Ann.
4045:Lie groups
4034:metrizable
3542:almost all
2209:dimensions
2156:continuous
2146:being any
1836:polynomial
1792:irrational
1645:to be the
933:such that
105:, ...) of
80:Definition
56:, ...) of
4633:MathWorld
4614:MathWorld
4435:(2006) .
4362:119558177
4227:0001-5962
4185:123470919
4030:separable
4000:μ
3997:⇒
3975:δ
3954:∑
3927:μ
3900:μ
3844:μ
3810:uniformly
3788:−
3777:−
3740:∩
3714:…
3679:∞
3676:→
3622:PV number
3492:is a set
3426:α
3400:α
3377:…
3365:α
3352:α
3345:α
3322:α
3299:…
3293:α
3284:α
3255:≥
3205:…
3092:∈
3064:…
2976:…
2879:−
2826:…
2718:α
2715:ℓ
2709:π
2698:−
2682:≤
2670:α
2667:ℓ
2661:π
2650:−
2640:α
2634:ℓ
2628:π
2617:−
2584:α
2581:ℓ
2575:π
2554:−
2536:∑
2517:α
2511:ℓ
2505:π
2489:−
2471:∑
2436:α
2433:ℓ
2427:π
2313:⋅
2310:ℓ
2304:π
2288:−
2270:∑
2254:∞
2251:→
2178:function
2158:function
2150:. By the
2115:ℓ
2109:π
2023:ℓ
2017:π
1989:∑
1973:∞
1970:→
1651:countable
1536:∑
1461:∑
1445:∞
1442:→
1431:≥
1385:∑
1369:∞
1366:→
1315:∫
1305:−
1239:∑
1223:∞
1220:→
1209:≤
1163:∑
1147:∞
1144:→
1093:∫
1083:−
1041:−
1032:ε
1029:≤
997:−
961:∫
854:−
843:−
753:∫
743:−
690:∑
674:∞
671:→
589:Suppose (
569:having a
503:−
492:−
483:−
455:∩
435:…
398:≤
392:≤
386:≤
236:−
225:−
188:∩
168:…
139:∞
136:→
4560:(2012).
4488:(1994).
4458:(1974).
4378:(1935),
4334:(1911),
4282:(2001).
4153:Weyl, H.
4055:See also
4038:standard
3085:′
3057:′
2991:, where
2862:′
2766:rational
2378:Suppose
1949:integers
1917:in 1948.
1757:Examples
1735:) or by
1657:is zero
1600:, where
567:function
115:interval
33:sequence
4021:In any
3596:> 1.
2395:(where
814:is the
4588:
4578:
4548:
4538:
4510:
4500:
4476:
4466:
4443:
4404:
4396:
4360:
4315:
4305:
4241:
4233:
4225:
4183:
4003:
3604:) or (
3556:zero.
3224:it is
886:linear
805:Proof
4358:S2CID
4287:(PDF)
4181:S2CID
4161:(PDF)
4077:Notes
4028:on a
4023:Borel
3620:is a
3612:) is
1904:, ...
1858:) is
1834:is a
1820:, ...
1653:, so
581:in a
565:is a
267:,...,
64:, or
4576:ISBN
4536:ISBN
4498:ISBN
4464:ISBN
4441:ISBN
4329:See
4303:ISBN
4223:ISSN
3403:>
1922:The
1900:, 11
1786:The
951:and
926:and
284:and
31:, a
4644:at
4586:Zbl
4546:Zbl
4508:Zbl
4474:Zbl
4402:Zbl
4394:JFM
4350:doi
4313:Zbl
4295:doi
4239:Zbl
4231:JFM
4213:doi
4173:doi
4040:.
3919:to
3812:in
3669:lim
3614:not
3500:in
3270:.
3165:.
2764:is
2363:, 3
2359:, 2
2244:lim
1963:lim
1951:ℓ,
1896:, 7
1892:, 5
1888:, 3
1860:not
1816:, 4
1812:, 3
1808:, 2
1804:0,
1717:of
1709:or
1516:of
1359:lim
1137:lim
888:in
664:lim
583:set
379:sup
288:.)
129:lim
27:In
4663::
4630:.
4611:.
4584:.
4574:.
4566:.
4544:.
4534:.
4506:.
4496:.
4472:.
4400:,
4388:,
4382:,
4356:,
4346:71
4344:,
4338:,
4311:.
4301:.
4271:^
4259:^
4237:,
4229:,
4221:,
4209:56
4207:,
4179:.
4169:77
4163:.
4106:^
4051:.
4032:,
3939::
3646:,
3639:,
3624:.
3532:.
3517:−
3488:A
3485:.
3467:−
3036::=
2866::=
2781:.
2779:jα
2454:,
2409:ℓα
2393:jα
2331:0.
2232:,
2162:.
2041:0.
1698:,
1691:,
1661:.
1604:,
1598:vi
1596:+
900:.
822:=
603:,
596:,
553:.
347:,
340:,
98:,
91:,
76:.
49:,
42:,
4648:.
4636:.
4617:.
4592:.
4552:.
4514:.
4480:.
4449:.
4390:2
4365:.
4352::
4319:.
4297::
4215::
4187:.
4175::
4006:.
3992:n
3984:k
3980:x
3969:n
3964:1
3961:=
3958:k
3880:)
3875:n
3871:x
3867:(
3847:)
3841:,
3838:X
3835:(
3814:k
3791:a
3785:b
3780:c
3774:d
3768:=
3763:n
3759:|
3755:]
3752:d
3749:,
3746:c
3743:[
3737:}
3731:n
3728:+
3725:k
3721:s
3717:,
3711:,
3706:1
3703:+
3700:k
3696:s
3691:{
3687:|
3673:n
3651:3
3648:s
3644:2
3641:s
3637:1
3634:s
3618:α
3610:α
3606:π
3602:e
3594:α
3590:α
3585:.
3583:α
3579:α
3575:n
3571:b
3566:n
3562:b
3550:α
3546:α
3529:n
3523:n
3519:s
3514:h
3512:+
3510:n
3506:s
3502:H
3498:h
3494:H
3482:n
3478:s
3473:n
3469:s
3464:h
3462:+
3460:n
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3380:)
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3361:,
3356:2
3348:,
3342:(
3302:)
3296:,
3290:2
3287:,
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3232:k
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3137:0
3134:[
3112:k
3108:]
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3098:0
3095:[
3089:)
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3075:n
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3000:b
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2947:(
2927:]
2924:1
2921:,
2918:0
2915:[
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2890:n
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2882:[
2874:n
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2858:n
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2829:)
2823:,
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2805:1
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2797:(
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2625:2
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2604:=
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2594:j
2589:)
2578:i
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2478:=
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2466:|
2430:i
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2401:ℓ
2397:j
2388:j
2384:a
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2301:2
2297:e
2291:1
2285:n
2280:0
2277:=
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2264:n
2261:1
2248:n
2230:Z
2226:R
2221:n
2217:v
2204:.
2184:∎
2180:f
2171:f
2167:f
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2144:f
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2123:,
2118:x
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2098:=
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2092:x
2089:(
2086:f
2038:=
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2027:a
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2010:e
2004:n
1999:1
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1983:n
1980:1
1967:n
1944:n
1940:a
1902:α
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1894:α
1890:α
1886:α
1884:2
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1806:α
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1606:v
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1590:f
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1556:f
1551:N
1546:1
1543:=
1540:n
1530:N
1527:1
1497:)
1492:n
1488:s
1484:(
1481:f
1476:N
1471:1
1468:=
1465:n
1455:N
1452:1
1439:N
1428:)
1423:n
1419:s
1415:(
1410:2
1406:f
1400:N
1395:1
1392:=
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1379:N
1376:1
1363:N
1355:=
1352:x
1349:d
1345:)
1342:x
1339:(
1334:2
1330:f
1324:b
1319:a
1308:a
1302:b
1298:1
1275:)
1270:n
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1262:(
1259:f
1254:N
1249:1
1246:=
1243:n
1233:N
1230:1
1217:N
1206:)
1201:n
1197:s
1193:(
1188:1
1184:f
1178:N
1173:1
1170:=
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1157:N
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1141:N
1133:=
1130:x
1127:d
1123:)
1120:x
1117:(
1112:1
1108:f
1102:b
1097:a
1086:a
1080:b
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1050:.
1047:)
1044:a
1038:b
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1016:)
1013:x
1010:(
1005:1
1001:f
994:)
991:x
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970:b
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935:f
931:2
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924:1
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879:f
863:.
857:a
851:b
846:c
840:d
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812:f
783:x
780:d
776:)
773:x
770:(
767:f
762:b
757:a
746:a
740:b
736:1
731:=
727:)
722:n
718:s
714:(
710:f
705:N
700:1
697:=
694:n
684:N
681:1
668:N
635:C
624:f
608:3
605:s
601:2
598:s
594:1
591:s
579:f
563:f
543:N
538:N
534:D
517:.
513:|
506:a
500:b
495:c
489:d
478:N
474:|
470:]
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464:,
461:c
458:[
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446:N
442:s
438:,
432:,
427:1
423:s
418:{
414:|
407:|
401:b
395:d
389:c
383:a
375:=
370:N
366:D
352:3
349:s
345:2
342:s
338:1
335:s
330:N
326:D
306:n
302:s
300:(
297:n
293:n
286:d
282:c
278:n
273:n
269:s
265:1
262:s
245:.
239:a
233:b
228:c
222:d
216:=
211:n
207:|
203:]
200:d
197:,
194:c
191:[
185:}
179:n
175:s
171:,
165:,
160:1
156:s
151:{
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133:n
103:3
100:s
96:2
93:s
89:1
86:s
54:3
51:s
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44:s
40:1
37:s
35:(
20:)
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