Henna Koivusalo

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

For any irrational cut-and-project setup, we demonstrate a natural infinite family of windows which gives rise to separated nets that are each bounded distance to a lattice. Our proof provides a new construction, using a sufficient condition of Rauzy, of an infinite family of non-trivial bounded remainder sets for any totally irrational toral rotation in any dimension.

Gaps problems and frequencies of patches in cut and project sets

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

Hausdorff dimension of affine random covering sets in torus

We calculate the almost sure Hausdorff dimension of the random covering set lim supn→∞(gn + ξn) in d-dimensional torus T, where the sets gn ⊂ T are parallelepipeds, or more generally, linear images of a set with nonempty interior, and ξn ∈ T are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

Projections of random covering sets

We show that, almost surely, the Hausdorff dimension

Dimensions of random affine code tree fractals

s_0

Hitting probabilities of random covering sets in tori and metric spaces

of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension

Self-affine sets with fibred tangents

k>s_0

Dimension of generic self-affine sets with holes

. The result holds for random covering sets with a generating sequence of ball-like sets, and is obtained by investigating orthogonal projections of a class of random Cantor sets.

A characterization of linearly repetitive cut and project sets

We study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random

Dimension of self‐affine sets for fixed translation vectors

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