Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An elementary approach to simplexes in thin subsets of Euclidean space   We prove that if the Hausdorff dimension of $E \subset {\Bbb R}^d$, $d \ge
3$, is greater than $\min \left\{ \frac{dk+1}{k+1}, \frac{d+k}{2} \right\},$
then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$, the set of
congruence classes of $k$-dimensional simplexes with vertices in $E$, is
positive. This improves the best bounds previously known, decreasing the
$\frac{d+k+1}{2}$ threshold obtained in Erdo\u{g}an-Hart-Iosevich (2012) to
$\frac{d+k}{2}$ via a different and conceptually simpler method. We also give a
simpler proof of the $d-\frac{d-1}{2d}$ threshold for $d$-dimensional simplexes
obtained in Greenleaf-Iosevich (2012), Grafakos-Greenleaf-Iosevich-Palsson
(2015).

12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
1
The categories of this paper are: cs
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin