On circle rotations and the shrinking target properties

By Jimmy Tseng. 2007

54 citations recorded by the source corpus.

We generalize the monotone shrinking target property (MSTP) to the $s$-exponent monotone shrinking target property ($s$MSTP) and give a necessary and sufficient condition for a circle rotation to have $s$MSTP.   Using another variant of MSTP, we obtain a new, very short, proof of a known result, which concerns the behavior of irrational rotations and implies a logarithm law similar to D. Sullivan's logarithm law for geodesics.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/34e4d2a01586590c40e2df1f94eab61edc25753a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Yann Bugeaud
2. Bao-Wei Wang
3. Jun Wu
4. Sanju Velani
5. Jian Xu
6. Bing Li
7. Manfred Einsiedler
8. Stephen Harrap

In Mathematical Physics

1. Yann Bugeaud
2. Bao-Wei Wang
3. Sanju Velani
4. Jun Wu
5. Jian Xu
6. Bing Li
7. S. Kristensen
8. Stephen Harrap

Citations in the corpus

• The shrinking target problem in the dynamical system of continued fractions (2014, background): “However, in some cases, the inequality in (1.2) which is stronger than mixing may not be satisfied, the conclusions of the dynamical Borel–Cantelli lemma still hold, for example the irrational rotations on circle for special sequences of measurable sets {Bn}n 1 (see Kim [22] and Tseng [33]).”
• Badly approximable systems of affine forms, fractals, and Schmidt games (2009, background): “The sets BadA(m,n) naturally arise as the complements of sets of toral translation vectors that satisfy certain shrinking target properties (see [21] and [3] for details).”
• On shrinking targets for Z^m actions on tori (2008, background): “Fayad [8℄, A.-H. Fan & J. Wu [7℄, Kim [11℄ and Tseng [22, 23℄ havebuilt upon the work of Kurzweil in various dire tions.”
• Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions (2014, background): “For the background and more results on shrinking target problems, the reader is referred to [1], [2], [3], [4], [8], [9], [10], [11], [12], [20], [21], [24], and [25] and the references quoted therein.”