Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Szeg\H{o}'s Condition on Compact subsets of $\mathbb{C}$   Let $K$ be a non-polar compact subset of $\mathbb{C}$ and $\mu_K$ be its
equilibrium measure. Let $\mu$ be a unit Borel measure supported on a compact
set which contains the support of $\mu_K$. We prove that a Szeg\H{o} condition
in terms of the Radon-Nikodym derivative of $\mu$ with respect to $\mu_K$
implies that $$\inf_n
\frac{\|P_n(\cdot;\mu)\|_{L^2(\mathbb{C};\mu)}}{\mathrm{Cap}(K)^n}>0.$$
  We show that
$\frac{\|P_n(\cdot;\mu_K)\|_{L^2(\mathbb{C};\mu_K)}}{\mathrm{Cap}(K)^n}\geq 1$
for any compact non-polar set $K$. We also prove that under an additional
assumption, unboundedness of the sequence
$\left(\frac{\|P_n(\cdot;\mu_K)\|_{L^2(\mathbb{C};\mu_K)}}{\mathrm{Cap}(K)^n}\right)$
implies that $K$ satisfies the Parreau-Widom condition.

10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph