On the dimension of $\alpha \beta$-sets

TL;DR

This paper proves the tightness of a known lower bound for the box dimension of lphaeta-sets and explores how removing small subsets affects their dimensions, revealing a disparity between box and Assouad dimensions.

Contribution

It establishes the exactness of the Feng-Xiong lower bound and analyzes the impact of small deletions on the dimensions of lphaeta-sets, highlighting differences between box and Assouad dimensions.

Findings

Feng-Xiong lower bound of 1/2 for box dimension is tight.

Removing small positive density sets can reduce box dimension to zero.

Assouad dimension cannot drop below 1/4 despite deletions.

Abstract

We show that the Feng-Xiong lower bound of $1/2$ for the box dimension of $\alpha\beta$-sets is tight. We also study how much of an $\alpha\beta$-orbit ``carries the dimension'': deleting an arbitararily small positive density set of times can cause the box dimension to drop to zero, but the Assouad dimension cannot drop below $1/4$.