Distinct gaps between fractional parts of sequences

Authors:
Marian Vâjâitu and Alexandru Zaharescu

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3447-3452

MSC (2000):
Primary 11K06, 11B05

DOI:
https://doi.org/10.1090/S0002-9939-02-06791-6

Published electronically:
July 15, 2002

MathSciNet review:
1918819

Full-text PDF

Abstract | References | Similar Articles | 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]**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, 267-307. 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, 1127-1172. MR**2001m:11112****[4]**van RAVENSTEIN, T.,*The three gap theorem (Steinhaus conjecture)*, J. Austral. Math. Soc. (Series A)**45**(1988), no.**3**, 360-370. MR**90a:11076****[5]**RUDNICK Z. and SARNAK P.,*The pair correlation function of fractional parts of polynomials*, Comm. Math. Phys.**194**(1998), 61-70. 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, 37-57. 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, 283-293. 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), 127-134.**[10]**SWIERCZKOWSKI, S.,*On successive settings of an arc on the circumference of a circle*, Fund. Math.**46**(1958), 187-189. MR**21:3404****[11]**WEYL, H.,*Uber die Gleichverteilung von Zahlen mod. Eins*. Math. Ann.**77**(1916 ), 313-352.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11K06,
11B05

Retrieve articles in all journals with MSC (2000): 11K06, 11B05

Additional Information

**Marian Vâjâitu**

Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania

Email:
mvajaitu@stoilow.imar.ro

**Alexandru Zaharescu**

Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania – and – Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, Illinois 61801

Email:
zaharesc@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06791-6

Received by editor(s):
February 7, 2001

Published electronically:
July 15, 2002

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 2002
American Mathematical Society