Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Infinite co-minimal pairs in the integers and integral lattices   Given two nonempty subsets $A, B$ of a group $G$, they are said to form a
co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any
$\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any
$\emptyset \neq B' \subsetneq B$. In this article, we show several new results
on co-minimal pairs in the integers and the integral lattices. We prove that
for any $d\geq 1$, the group $\mathbb{Z}^{2d}$ admits infinitely many
automorphisms such that for each such automorphism $\sigma$, there exists a
subset $A$ of $\mathbb{Z}^{2d}$ such that $A$ and $\sigma(A)$ form a co-minimal
pair. The existence and construction of co-minimal pairs in the integers with
both the subsets $A$ and $B$ ($A\neq B$) of infinite cardinality was unknown.
We show that such pairs exist and explicitly construct these pairs satisfying a
number of algebraic properties.

32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat