Tiling of polyhedral sets

TL;DR

This paper classifies all polyhedral sets that can be partitioned into self-affine tiles, revealing an infinite variety of such sets in Euclidean space and exploring their applications in approximation theory and functional analysis.

Contribution

It characterizes all polyhedral sets admitting self-affine tilings and identifies an infinite family of non-equivalent examples, especially focusing on integer affine similarities.

Findings

Existence of infinitely many non-affinely equivalent polyhedral self-affine tiles in R^d.

Identification of special cases with integer matrices and translation vectors.

Applications demonstrated in approximation theory and functional analysis.

Abstract

A self-affine tiling of a compact set G of positive Lebesgue measure is its partition to parallel shifts of a compact set which is affinely similar to G. We find all polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings. It is shown that in R^d there exist an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to an important particular case when the matrix of affine similarity and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed.