Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long Arithmetic Progressions in Sets with Small Sumset   Let $A, B\subseteq \mathbb{Z}$ be finite, nonempty subsets with $\min A=\min
B=0$, and let $$\delta(A,B)={\begin{array}{ll}
  1 & \hbox{if} A\subseteq B,
  0 & \hbox{otherwise.}
  If $\max B\leq \max A\leq |A|+|B|-3$ and \label{one}|A+B|\leq
|A|+2|B|-3-\delta(A,B), then we show $A+B$ contains an arithmetic progression
with difference 1 and length $|A|+|B|-1$.
  As a corollary, if \eqref{one} holds, $\max(B)\leq \max(A)$ and either
$\gcd(A)=1$ or else $\gcd(A+B)=1$ and $|A+B|\leq 2|A|+|B|-3$, then $A+B$
contains an arithmetic progression with difference 1 and length $|A|+|B|-1$.

34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
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The categories of this paper are: eess
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th