On Lebesgue measure of integral self-affine sets

TL;DR

This paper introduces an effective algorithm for computing the Lebesgue measure of self-affine sets, their intersections with translations, and intersections between different self-affine sets, advancing understanding of their measure-theoretic properties.

Contribution

It provides a novel, practical algorithm to calculate measures and intersections of integral self-affine sets, which was previously difficult to determine explicitly.

Findings

Algorithm successfully computes Lebesgue measure of self-affine sets.

It determines measures of intersections with translations.

It calculates measures of intersections between different self-affine sets.

Abstract

Let $A$ be an expanding integer $n\times n$ matrix and $D$ be a finite subset of $Z^n$. The self-affine set $T=T(A,D)$ is the unique compact set satisfying the equality $A(T)=\cup_{d\in D} (T+d)$. We present an effective algorithm to compute the Lebesgue measure of the self-affine set $T$, the measure of intersection $T\cap (T+u)$ for $u\in Z^n$, and the measure of intersection of self-affine sets $T(A,D_1)\cap T(A,D_2)$ for different sets $D_1,D_2\subset Z^n$.