## On the limit points of $(a_n\xi )_{n=1}^{\infty }$ mod $1$ for slowly increasing integer sequences $(a_n)_{n=1}^{\infty }$

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- by Artūras Dubickas PDF
- Proc. Amer. Math. Soc.
**137**(2009), 449-456 Request permission

## Abstract:

In this paper, we are interested in sequences of positive integers $(a_n)_{n=1}^{\infty }$ such that the sequence of fractional parts $\{a_n\xi \}_{n=1}^{\infty }$ has only finitely many limit points for at least one real irrational number $\xi .$ We prove that, for any sequence of positive numbers $(g_n)_{n=1}^{\infty }$ satisfying $g_n \geq 1$ and $\lim _{n\to \infty } g_n=\infty$ and any real quadratic algebraic number $\alpha ,$ there is an increasing sequence of positive integers $(a_n)_{n=1}^{\infty }$ such that $a_n \leq n g_n$ for every $n \in \mathbb {N}$ and $\lim _{n\to \infty }\{a_n \alpha \} = 0.$ The above bound on $a_n$ is best possible in the sense that the condition $\lim _{n\to \infty } g_n=\infty$ cannot be replaced by a weaker condition. More precisely, we show that if $(a_n)_{n=1}^{\infty }$ is an increasing sequence of positive integers satisfying $\liminf _{n\to \infty } a_n/n<\infty$ and $\xi$ is a real irrational number, then the sequence of fractional parts $\{a_n \xi \}_{n=1}^{\infty }$ has infinitely many limit points.## References

- Daniel Berend,
*Multi-invariant sets on tori*, Trans. Amer. Math. Soc.**280**(1983), no. 2, 509–532. MR**716835**, DOI 10.1090/S0002-9947-1983-0716835-6 - Daniel Berend,
*Multi-invariant sets on compact abelian groups*, Trans. Amer. Math. Soc.**286**(1984), no. 2, 505–535. MR**760973**, DOI 10.1090/S0002-9947-1984-0760973-X - Daniel Berend,
*Actions of sets of integers on irrationals*, Acta Arith.**48**(1987), no. 3, 275–306. MR**921090**, DOI 10.4064/aa-48-3-275-306 - Michael D. Boshernitzan,
*Elementary proof of Furstenberg’s Diophantine result*, Proc. Amer. Math. Soc.**122**(1994), no. 1, 67–70. MR**1195714**, DOI 10.1090/S0002-9939-1994-1195714-X - B. de Mathan,
*Numbers contravening a condition in density modulo $1$*, Acta Math. Acad. Sci. Hungar.**36**(1980), no. 3-4, 237–241 (1981). MR**612195**, DOI 10.1007/BF01898138 - Harry Furstenberg,
*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**213508**, DOI 10.1007/BF01692494 - H. Halberstam and K. F. Roth,
*Sequences. Vol. I*, Clarendon Press, Oxford, 1966. MR**0210679** - A. Khintchine,
*Über eine Klasse linearer diophantischer Approximationen,*Rend. Circ. Mat. Palermo,**50**(1926), 170–195. - Bryna Kra,
*A generalization of Furstenberg’s Diophantine theorem*, Proc. Amer. Math. Soc.**127**(1999), no. 7, 1951–1956. MR**1487320**, DOI 10.1090/S0002-9939-99-04742-5 - David Meiri,
*Entropy and uniform distribution of orbits in $\textbf {T}^d$*, Israel J. Math.**105**(1998), 155–183. MR**1639747**, DOI 10.1007/BF02780327 - C. Pisot,
*La répartition modulo 1 et les nombres algébriques,*Ann. Scuola Norm. Sup. Pisa,**7**(1938), 204–248. - A. D. Pollington,
*On the density of sequence $\{n_{k}\xi \}$*, Illinois J. Math.**23**(1979), no. 4, 511–515. MR**540398** - Oto Strauch and Štefan Porubský,
*Distribution of sequences: a sampler*, Schriftenreihe der Slowakischen Akademie der Wissenschaften [Series of the Slovak Academy of Sciences], vol. 1, Peter Lang, Frankfurt am Main, 2005. MR**2290224** - Roman Urban,
*Sequences of algebraic numbers and density modulo 1*, Publ. Math. Debrecen**72**(2008), no. 1-2, 141–154. MR**2376865** - T. Vijayaraghavan,
*On the fractional parts of the powers of a number. I*, J. London Math. Soc.**15**(1940), 159–160. MR**2326**, DOI 10.1112/jlms/s1-15.2.159 - Hermann Weyl,
*Über die Gleichverteilung von Zahlen mod. Eins*, Math. Ann.**77**(1916), no. 3, 313–352 (German). MR**1511862**, DOI 10.1007/BF01475864

## Additional Information

**Artūras Dubickas**- Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
- Email: arturas.dubickas@mif.vu.lt
- Received by editor(s): December 17, 2007
- Received by editor(s) in revised form: January 19, 2008
- Published electronically: August 4, 2008
- Communicated by: Ken Ono
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 449-456 - MSC (2000): Primary 11B05, 11B37, 11J71, 11R11
- DOI: https://doi.org/10.1090/S0002-9939-08-09491-4
- MathSciNet review: 2448563