Related papers: Differentiability of functions in the Zygmund clas…
We consider the space of real-valued continuously differentiable functions on a compact subset of a euclidean space. We characterize the completeness of this space and prove that the space of restrictions of continuously differentiable…
Eremenko and Lyubich proved that an entire function whose set of singular values is bounded is expanding at points where its image has large modulus. These expansion properties have been at the centre of the subsequent study of this class…
The classical Hausdorff dimension of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension which is non-trivial. It turns out that a finite bound for the finite Hausdorff dimension…
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…
In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional…
The Hausdorff dimension of the set of points that are covered infinitely many times by a sequence of randomly distributed balls in the unit cube can be expressed in terms of the sizes of the balls. This note presents a new proof of the…
We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of $\varepsilon$-Dirichlet improvable vectors in this setting. This is a function field analogue of the…
The set B of geodesic rays avoiding a suitable obstacle in a complete negatively curved Riemannian manifold determines a spectrum S. While various properties of this spectrum are known, we define and study dimension functions on S in terms…
We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in…
We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical…
We introduce a higher dimensional quasiregular map analogous to the trigonometric functions and we use the dynamics of this map to define, for d>1, a partition of d-dimensional Euclidean space into curves tending to infinity such that two…
We prove that for $d\ge 2,\, k\ge 2$, if the Hausdorff dimension of a compact set $E\subset \mathbb{R}^d$ is greater than $\frac{d^2}{2d-1}$, then, for any given $r > 0$, there exist $(x^1, \dots, x^{k+1})\in E^{k+1}$, $(y^1, \dots,…
We study some forward invariant sets appearing in the dynamics of the exponential family. We prove that the Hausdorff dimension of the sets under consideration is not larger than $1$. This allows us to prove, as a consequence, a result for…
We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a…
This paper examines level sets of two families of continuous, nowhere differentiable functions (one a subfamily of the other) defined in terms of the "tent map". The well-known Takagi function is a special case. Sharp upper bounds are given…
On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…
In this work we found order estimates for the upper bounds of the deviations of analogue of Zigmund's sums on the classes of $(\psi;\beta)$-differentiable functions in the metrics of generalized Lebesgue spaces with variable exponent.
We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parametrizing matrix. This provides evidence towards the conjecture that all affine subspaces…
We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for…
We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. The proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates, factorization…