Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some problems on the boundary of fractal geometry and additive
  combinatorics   This paper is an exposition, with some new applications, of our results on
the growth of entropy of convolutions. We explain the main result on
$\mathbb{R}$, and derive, via a linearization argument, an analogous result for
the action of the affine group on $\mathbb{R}$. We also develop versions of the
results for entropy dimension and Hausdorff dimension. The method is applied to
two problems on the border of fractal geometry and additive combinatorics.
First, we consider attractors $X$ of compact families $\Phi$ of similarities of
$\mathbb{R}$. We conjecture that if $\Phi$ is uncountable and $X$ is not a
singleton (equivalently, $\Phi$ is not contained in a 1-parameter semigroup)
then $\dim X=1$. We show that this would follow from the classical overlaps
conjecture for self-similar sets, and unconditionally we show that if $X$ is
not a point and $\dim\Phi>0$ then $\dim X=1$. Second, we study a problem due to
Shmerkin and Keleti, who have asked how small a set $\emptyset\neq
Y\subseteq\mathbb{R}$ can be if at every point it contains a scaled copy of the
middle-third Cantor set $K$. Such a set must have dimension at least $\dim K$
and we show that its dimension is at least $\dim K+\delta$ for some constant
$\delta>0$.

14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
23
The categories of this paper are: solv-int