Related papers: On shrinking targets for Z^m actions on tori

We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a…

In this paper, we prove that if $S\subseteq\mathbb{R}^d$ is hyperplane absolute winning on a closed hyperplane diffuse set $L\subseteq\mathbb{R}^d$, then $\mathrm{dim}_H S\cap K=\mathrm{dim}_H K$ for any irreducible self-conformal set…

Suppose $A \in \mathbb{R}^{n \times n}$ is invertible and we are looking for the solution of $Ax = b$. Given an initial guess $x_1 \in \mathbb{R}$, we show that by reflecting through hyperplanes generated by the rows of $A$, we can generate…

We address the problem of the continuum limit for a system of Hausdorff lattices (namely lattices of isolated points) approximating a topological space $M$. The correct framework is that of projective systems. The projective limit is a…

The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal…

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the…

For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…

Let $X$ be a compact metric space which is locally absolutely retract and let $\phi: C(X)\to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${\rm dim}Y\le 2.$ It is proved that there exists a sequence of $n$…

We investigate the set of $x \in S^1$ such that for every positive integer $N$, the first $N$ points in the orbit of $x$ under rotation by irrational $\theta$ contain at least as many values in the interval $[0,1/2]$ as in the complement.…

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and…

Let $(R, \mathfrak{m})$ be a regular local ring of characteristic $p > 0$. Among all proper ideals $\mathfrak{a}\subseteq R$ with a fixed order of vanishing $\text{ord}_{\mathfrak{m}}(\mathfrak{a})$, we classify the ideals for which the…

We prove a version of the Erd\H{o}s--Beck Theorem from discrete geometry for fractal sets in all dimensions. More precisely, let $X\subset \mathbb{R}^n$ Borel and $k \in [0, n-1]$ be an integer. Let $\dim (X \setminus H) = \dim X$ for every…

Recent findings by Jahn, T. Ullrich, Voigtlaender [10] relate non-linear sampling numbers for the square norm to quantities involving trigonometric best $m-$term approximation errors in the uniform norm. Here we establish new results for…

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which…

Consider a real matrix $\Theta$ consisting of rows $(\theta_{i,1},\ldots,\theta_{i,n})$, for $1\leq i\leq m$. The problem of making the system linear forms $x_{1}\theta_{i,1}+\cdots+x_{n}\theta_{i,n}-y_{i}$ for integers $x_{j},y_{i}$ small…

It is well known due to Jarnik that the set Bad of badly approximable numbers is of Hausdorff-dimension one. If Bad(c) denotes the subset of x in Bad for which the approximation constant c > c(x), then Jarnik was in fact more precise and…

Let $A \in \mathbb{R}^{n \times n}$ be invertible, $x \in \mathbb{R}^n$ unknown and $b =Ax $ given. We are interested in approximate solutions: vectors $y \in \mathbb{R}^n$ such that $\|Ay - b\|$ is small. We prove that for all $0<…

The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important…

Consider the integer best approximations of a linear form in $n\ge 2$ real variables. While it is well-known that any tail of this sequence always spans a lattice is sharp for any $n\ge 2$. In this paper, we determine the exact Hausdorff…

This paper considers the non-convex problem of finding the nearest Metzler matrix to a given possibly unstable matrix. Linear systems whose state vector evolves according to a Metzler matrix have many desirable properties in analysis and…