Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Singular Vectors on Fractals and Projections of Self-similar Measures   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.

36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
3
The categories of this paper are: physics