# A self-similar measure with dense rotations, singular projections and discrete slices

## TL;DR

This paper constructs a planar self-similar measure with dense rotations and high dimension, demonstrating that in certain directions, projections are singular and slices are discrete, challenging typical dimension conservation expectations.

## Contribution

It introduces a specific self-similar measure with dense rotations and shows that dimension conservation fails in a residual set of directions with singular projections and discrete slices.

## Key findings

- Existence of directions with singular projections
- Residual set of such directions
- Typical slices are discrete

## Abstract

We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than $1$, such that there exist lines for which dimension conservation does not hold and the projection of the measure is singular. In fact, the set of such directions is residual and the typical slices of the measure, perpendicular to these directions, are discrete.