Distinct gaps between fractional parts of sequences

Vâjâitu, Marian; Zaharescu, Alexandru

doi:10.1090/S0002-9939-02-06791-6

Distinct gaps between fractional parts of sequences
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by Marian Vâjâitu and Alexandru Zaharescu PDF

Proc. Amer. Math. Soc. 130 (2002), 3447-3452 Request permission

Abstract:

Let $\alpha$ be a real number, $N$ a positive integer and $\mathcal N$ a subset of $\{0,1,2,\dots ,N\}$. We give an upper bound for the number of distinct lengths of gaps between the fractional parts $\{ n\alpha \},\;n\in \mathcal N$.

References

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
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, DOI 10.1215/S0012-7094-00-10524-8
Jens Marklof, The $n$-point 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, DOI 10.1017/S0143385700000626
Tony van Ravenstein, The three gap theorem (Steinhaus conjecture), J. Austral. Math. Soc. Ser. A 45 (1988), no. 3, 360–370. MR 957201
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, DOI 10.1007/s002200050348
Zeév Rudnick, Peter Sarnak, and Alexandru Zaharescu, The distribution of spacings between the fractional parts of $n^2\alpha$, Invent. Math. 145 (2001), no. 1, 37–57. MR 1839285, DOI 10.1007/s002220100141
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, DOI 10.4064/aa-89-3-283-293
Wolfgang M. Schmidt, Small fractional parts of polynomials, Regional Conference Series in Mathematics, No. 32, American Mathematical Society, Providence, R.I., 1977. MR 0457360
SÓS, V. T., On the distribution $\mod 1$ of the sequence n$\alpha ,$ Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1 (1958), 127–134.
S. Świerczkowski, On successive settings of an arc on the circumference of a circle, Fund. Math. 46 (1959), 187–189. MR 104651, DOI 10.4064/fm-46-2-187-189
WEYL, H., Uber die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916 ), 313-352.