Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Answer to a question of Kolmogorov   More than 80 years ago Kolmogorov asked the following question. Let
$E\subseteq \mathbb{R}^{2}$ be a measurable set with $\lambda^{2}(E)<\infty$,
where $\lambda^2$ denotes the two-dimensional Lebesgue measure. Does there
exist for every $\varepsilon>0$ a contraction $f\colon E\to \mathbb{R}^{2}$
such that $\lambda^{2}(f(E))\geq \lambda^{2}(E)-\varepsilon$ and $f(E)$ is a
polygon? We answer this question in the negative by constructing a bounded,
simply connected open counterexample. Our construction can easily be modified
to yield the analogous result in higher dimensions.

8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci