Related papers: Accumulation points of normalized approximations

In this paper we construct a new family of sets based on Diophantine approximation in the Euclidean space, and consider their applications in several problems in harmonic analysis. Our first application is on the Hausdorff dimension of our…

We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khinchin and Jarnik theorems. In full generality our results establish…

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with…

Let $k$ be a number field and $K$ a finite extension of $k$. We count points of bounded height in projective space over the field $K$ generating the extension $K/k$. As the height gets large we derive asymptotic estimates with a…

For any $\beta>1$, let $T_\beta$ be the classical $\beta$-transformations. Fix $x_0\in[0,1]$ and a nonnegative real number $\hat{v}$, we compute the Hausdorff dimension of the set of real numbers $x\in[0,1]$ with the property that, for…

We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points…

Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation with a given bound. We study uniform Diophantine approximation properties on the Hecke group $\mathbf…

Let $\Gamma = Z A +Z^n$ be a dense subgroup with rank $n+1$ in $R^n$ and let $\omega(A)$ denote the exponent of uniform simultaneous rational approximation to the point $A$. We show that for any real number $v\ge \omega(A)$, the Hausdorff…

In this paper we show that the set of accumulation points of generalized log canonical thresholds for certain DCC sets comes from the set of generalized log canonical thresholds of dimension $1$ less of the same DCC sets.

Let g be a nonconstant rational map from the projective line to itself that has degree greater than one and is defined over a number field. The map g gives rise to generalized Mahler measures for polynomials in one variable. We use…

We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of…

We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest…

Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is…

We show that Mahler's classification of real numbers $\zeta$ with respect to the growth of the sequence $(w_{n}(\zeta))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence…

Let $\theta\in\mathbb{R}^d$. We associate three objects to each approximation $(p,q)\in \mathbb{Z}^d\times \mathbb{N}$ of $\theta$: the projection of the lattice $\mathbb{Z}^{d+1}$ to the hyperplane of the first $d$ coordinates along the…

The $k$ multiplicative and $k$ additive compounds of a matrix play an important role in geometry, multi-linear algebra, the asymptotic analysis of nonlinear dynamical systems, and in bounding the Hausdorff dimension of fractal sets. These…

Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) =…

The paper is devoted to an algebraic analogue of a geometric approach to the classical notion of complex dilatation suggested in the paper arXiv:1701.06259 [math.CV] by the author. At the same time it provides an invariant version of this…

We quantify the density of rational points in the unit sphere $S^n$, proving analogues of the classical theorems on the embedding of $\q^n$ into $\r^n$. Specifically, we prove a Dirichlet theorem stating that every point $\alpha \in S^n$ is…

We investigate the Hausdorff dimension of level sets defined by digit growth rates in $\theta$-expansions, a generalization of regular continued fractions. For any $\alpha \geq 0$, we prove that the set \[ E_\theta(\alpha) = \left\{ x \in…