Distinct gaps between fractional parts of sequences
Authors:
Marian Vâjâitu and Alexandru Zaharescu
Journal:
Proc. Amer. Math. Soc. 130 (2002), 34473452
MSC (2000):
Primary 11K06, 11B05
Published electronically:
July 15, 2002
MathSciNet review:
1918819
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: Let be a real number, a positive integer and a subset of . We give an upper bound for the number of distinct lengths of gaps between the fractional parts .
 [1]
R.
C. Baker, Diophantine inequalities, London Mathematical
Society Monographs. New Series, vol. 1, The Clarendon Press, Oxford
University Press, New York, 1986. Oxford Science Publications. MR 865981
(88f:11021)
 [2]
Florin
P. Boca and Alexandru
Zaharescu, Pair correlation of values of rational functions (mod
𝑝), Duke Math. J. 105 (2000), no. 2,
267–307. MR 1793613
(2001j:11065), http://dx.doi.org/10.1215/S0012709400105248
 [3]
Jens
Marklof, The 𝑛point correlations between values of a
linear form, Ergodic Theory Dynam. Systems 20 (2000),
no. 4, 1127–1172. With an appendix by Zeév Rudnick. MR 1779397
(2001m:11112), http://dx.doi.org/10.1017/S0143385700000626
 [4]
Tony
van Ravenstein, The three gap theorem (Steinhaus conjecture),
J. Austral. Math. Soc. Ser. A 45 (1988), no. 3,
360–370. MR
957201 (90a:11076)
 [5]
Zeév
Rudnick and Peter
Sarnak, The pair correlation function of fractional parts of
polynomials, Comm. Math. Phys. 194 (1998),
no. 1, 61–70. MR 1628282
(99g:11088), http://dx.doi.org/10.1007/s002200050348
 [6]
Zeév
Rudnick, Peter
Sarnak, and Alexandru
Zaharescu, The distribution of spacings between the fractional
parts of 𝑛²𝛼, Invent. Math. 145
(2001), no. 1, 37–57. MR 1839285
(2002e:11093), http://dx.doi.org/10.1007/s002220100141
 [7]
Zeév
Rudnick and Alexandru
Zaharescu, A metric result on the pair correlation of fractional
parts of sequences, Acta Arith. 89 (1999),
no. 3, 283–293. MR 1691857
(2000h:11083)
 [8]
Wolfgang
M. Schmidt, Small fractional parts of polynomials, American
Mathematical Society, Providence, R.I., 1977. Regional Conference Series in
Mathematics, No. 32. MR 0457360
(56 #15568)
 [9]
SÓS, V. T., On the distribution of the sequence n Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1 (1958), 127134.
 [10]
S.
Świerczkowski, On successive settings of an arc on the
circumference of a circle, Fund. Math. 46 (1959),
187–189. MR 0104651
(21 #3404)
 [11]
WEYL, H., Uber die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916 ), 313352.
 [1]
 BAKER R.C., Diophantine inequalities, London Math. Soc. Monographs. New Series, 1. The Clarendon Press, Oxford Univ. Press, New York, 1986. MR 88f:11021
 [2]
 BOCA F. and ZAHARESCU A., Pair correlation of values of rational functions , Duke Math. Journal 105 (2000), no 2, 267307. MR 2001j:11065
 [3]
 MARKLOF, J., The point correlations between values of a linear form. With an appendix by Zeev Rudnick, Ergodic Theory Dynam. Systems 20 (2000), no. 4, 11271172. MR 2001m:11112
 [4]
 van RAVENSTEIN, T., The three gap theorem (Steinhaus conjecture), J. Austral. Math. Soc. (Series A) 45 (1988), no. 3, 360370. MR 90a:11076
 [5]
 RUDNICK Z. and SARNAK P., The pair correlation function of fractional parts of polynomials, Comm. Math. Phys. 194 (1998), 6170. MR 99g:11088
 [6]
 RUDNICK Z., SARNAK P. and ZAHARESCU A., The distribution of spacings between the fractional parts of , Invent. Math. 145 (2001), no 1, 3757. MR 2002e:11093
 [7]
 RUDNICK Z. and ZAHARESCU A., A metric result on the pair correlation of fractional parts of sequences, Acta Arith. 89 (1999), no.3, 283293. MR 2000h:11083
 [8]
 SCHMIDT W.M., Small fractional parts of polynomials, Regional Conference Series in Mathematics, No. 32, Amer. Math. Soc., Providence, RI, 1977. MR 56:15568
 [9]
 SÓS, V. T., On the distribution of the sequence n Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1 (1958), 127134.
 [10]
 SWIERCZKOWSKI, S., On successive settings of an arc on the circumference of a circle, Fund. Math. 46 (1958), 187189. MR 21:3404
 [11]
 WEYL, H., Uber die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916 ), 313352.
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Additional Information
Marian Vâjâitu
Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1764, Bucharest 70700, Romania
Email:
mvajaitu@stoilow.imar.ro
Alexandru Zaharescu
Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1764, Bucharest 70700, Romania – and – Department of Mathematics, University of Illinois at UrbanaChampaign, Altgeld Hall, 1409 W. Green Street, Urbana, Illinois 61801
Email:
zaharesc@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S0002993902067916
PII:
S 00029939(02)067916
Received by editor(s):
February 7, 2001
Published electronically:
July 15, 2002
Communicated by:
David E. Rohrlich
Article copyright:
© Copyright 2002
American Mathematical Society
