Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of Bad_A with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.
We study the shrinking target problem for actions of semisimple groups on homogeneous spaces, with applications to logarithm laws and Diophantine approximation.
We show that a translation of vector V on the torus has the monotone shrinking target property if and only if the vector V is of constant type. Then, using reparametrizations of linear flows, we show that there exist area preserving real analytic maps of the three torus that are mixing of all orders and do not enjoy the monotone shrinking target property.
We generalize the monotone shrinking target property (MSTP) to the s-exponent monotone shrinking target property (sMSTP) and give a necessary and sufficient condition for a circle rotation to have sMSTP. 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.
We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation, and probability theory.
We discuss the shrinking target property of irrational rotations. We obtain the condition of an irrational $θ$ and monotone increasing $φ(n)$ such that $$ \liminf_{n \to \infty} n φ(n) \| nθ- s \| = 0 \text{ for almost every } s. $$ We also consider the class of irrationals for which the limit inferior is 0 for every monotone increasing $φ(n)$ such that $\sum_n 1/(nφ(n))$ diverges.
This paper proves shrinking target results for IETs. Let {a_1\geq a_2 \geq...} be a sequence of positive real numbers with divergent sum. Then for almost every IET T, the limsup of B(T^ix,a_i) has full Lebesgue measure (where B(z, e) is the open ball around z of radius e). Related results are established including the analogous result for geodesic flows on a translation surface.
In this article, we study the Hausdorff measure of shrinking target sets on self-conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our "shrinking targets". Our main result complements earlier work o…
We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the non-autonomous dynamical system on the interval $[0,1)$, viewed as $\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilizing our theorem. In particular, we provide a Diophantine approximation interpretation of our scheme.
We study shrinking target problems and the set $\mathcal{E}_{\text{ah}}$ of eventually always hitting points. These are the points whose first $n$ iterates will never have empty intersection with the $n$-th target for sufficiently large $n$. We derive necessary and sufficient conditions on the shrinking rate of the targets for $\mathcal{E}_{\text{ah}}$ to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville-Pomeau map. We also obtain results for the Gauss map and correspondingly for the maximal digits in continued fract…
We consider the two dimensional shrinking target problem in the beta dynamical system for general $β>1$ and with the general error of approximations. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in[0,1]$, define the shrinking target set $$ E(T_β, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{ll} |T_β^{n}x-x_{0}|<e^{-S_nf(x)}\\ [1ex] |T_β^{n}y-y_{0}|< e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $S_nf(x)=\sum_{j=0}^{n-1}f(T_β^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to…
We give an integrability criterion on a real-valued non-increasing function $ψ$ guaranteeing that for almost all (or almost no) pairs $(A, \textbf{b})$, where $A$ is a real $m\times n$ matrix and $\textbf{b} \in \mathbb{R}^m$, the system $\|A \textbf{q}+\textbf{b}-\textbf{p}\|^m< ψ({T})$, $\|\textbf{q}\|^n<{T}$ is solvable in $\textbf{p} \in \mathbb{Z}^m$, $\textbf{q} \in \mathbb{Z}^n$ for all sufficiently large $T$. The proof consists of a reduction to a shrinking target problem on the space of grids in $\mathbb{R}^{m+n}$. We also comment on the homogeneous counterpart to this problem, whose …
Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of $T^{-1}$ as $T \rightarrow \infty$. We show that a closed horocycle whose length $\ell$ goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate $T^{-1}$ provided that $T/\ell \rightarrow 0$ and, for any $δ>0$, the segment remains larger than $\max\left\{T^{-1/6},\left(T/\ell\right)^{1/2}\right\}\left(T/\ell\right)^{-δ}$. We also have an e…
Relying on the classical second moment formula of Rogers we give an effective asymptotic formula for the number of integer vectors $v$ in a ball of radius $t$, with value $Q(v)$ in a shrinking interval of size $t^{-κ}$, that is valid for almost all indefinite quadratic forms in $n$ variables for any $κ<n-2$. This implies in particular, the existence of such integer solutions establishing the prediction made by Ghosh Gorodnik and Nevo. We also obtain similar results for random polynomials of higher degree.
In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero-one laws for the two problems.
