Arithmetic structure of the exceptional set of projections

TL;DR

This paper investigates the algebraic structure of the set of projections that preserve the box dimension of a set, revealing it is either trivial or a subfield, and constructs examples with specific dimensional properties.

Contribution

It establishes a dichotomy for the structure of the exceptional set of projections and provides new constructions of sets with controlled dimensional behavior under projections.

Findings

The exceptional set of projections is either {0} or a subfield of a9a9.

The property does not extend to Hausdorff or lower box dimensions.

Constructed Ahlfors s-regular sets with projections having strictly smaller upper box dimension.

Abstract

We study the arithmetic structure of the exceptional set of projections. For any bounded subset $E\subset \mathbb{R}^d$, let $$ \Omega=\{\xi\in \mathbb{R}: \dim_B(E+\xi E)=\dim_B E\}. $$ We prove that either $\Omega=\{0\}$ or $\Omega$ is a subfield of $\mathbb{R}$. We show that in general the statement does not hold for Hausdorff dimension and lower box dimension. Moreover, for any $s\in (0, 1]$ and a sequence $(r_k) \subset \mathbb{R}$, we construct a Ahlfors $s$-regular set $E\subset \mathbb{R}^2$ such that for any $r_k, k\in \mathbb{N}$, we have \[ \overline{\dim}_B \, \{x+r_k\, y: (x, y)\in E\} <s. \]