Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Mattila-Sj\"olin distance theorem for product sets   Let $A$ be a compact set in $\mathbb{R}$, and $E=A^d\subset \mathbb{R}^d$. We
know from the Mattila-Sj\"olin's theorem if $\dim_H(A)>\frac{d+1}{2d}$, then
the distance set $\Delta(E)$ has non-empty interior. In this paper, we show
that the threshold $\frac{d+1}{2d}$ can be improved whenever $d\ge 5$.

28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess