# Singular Vectors on Fractals and Projections of Self-similar Measures

## TL;DR

This paper establishes an upper bound on the Hausdorff dimension of singular vectors on self-similar fractals, linking Diophantine approximation properties with fractal geometry, and applies the results to translation flows on flat surfaces.

## Contribution

It provides an optimal upper bound on the Hausdorff dimension of singular vectors on self-similar fractals, extending previous results and addressing open questions in the field.

## Key findings

- Upper bound matches known exact dimensions for trivial fractals.
- For specific fractals like Cantor sets, the bound is 2/3 of the fractal's dimension.
- The method applies to translation flows, showing non-uniquely ergodic directions have dimension at most 1/2 of the fractal.

## Abstract

Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular vectors lying on self-similar fractals in $\mathbb{R}^d$ satisfying the open set condition. The bound is in terms of quantities which are closely tied to Frostman exponents of projections of the Hausdorff measure supported on the fractal. Our bound is optimal in the sense that it agrees with the exact dimension of singular vectors obtained by Cheung and Chevallier when the fractal is trivial (i.e. has non-empty interior). As a corollary, we show that if the fractal is the product of $2$ copies of Cantor's middle thirds set or the attractor of a planar homogeneous irrational IFS, then the upper bound is $2/3$ the dimension of the fractal. This addresses the upper bound part of a question raised by Bugeaud, Cheung and Chevallier. We apply our method in the setting of translation flows on flat surfaces to show that the dimension of non-uniquely ergodic directions belonging to a fractal is at most $1/2$ the dimension of the fractal.