Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sums and differences of correlated random sets   Many fundamental questions in additive number theory (such as Goldbach's
conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be
expressed in the language of sum and difference sets. As a typical pair of
elements contributes one sum and two differences, we expect that $|A-A| >
|A+A|$ for a finite set $A$. However, in 2006 Martin and O'Bryant showed that a
positive proportion of subsets of $\{0, \dots, n\}$ are sum-dominant, and Zhao
later showed that this proportion converges to a positive limit as $n \to
\infty$. Related problems, such as constructing explicit families of
sum-dominant sets, computing the value of the limiting proportion, and
investigating the behavior as the probability of including a given element in
$A$ to go to zero, have been analyzed extensively.
  We consider many of these problems in a more general setting. Instead of just
one set $A$, we study sums and differences of pairs of \emph{correlated} sets
$(A,B)$. Specifically, we place each element $a \in \{0,\dots, n\}$ in $A$ with
probability $p$, while $a$ goes in $B$ with probability $\rho_1$ if $a \in A$
and probability $\rho_2$ if $a \not \in A$. If $|A+B| > |(A-B) \cup (B-A)|$, we
call the pair $(A,B)$ a \emph{sum-dominant $(p,\rho_1, \rho_2)$-pair}. We prove
that for any fixed $\vec{\rho}=(p, \rho_1, \rho_2)$ in $(0,1)^3$, $(A,B)$ is a
sum-dominant $(p,\rho_1, \rho_2)$-pair with positive probability, and show that
this probability approaches a limit $P(\vec{\rho})$. Furthermore, we show that
the limit function $P(\vec{\rho})$ is continuous. We also investigate what
happens as $p$ decays with $n$, generalizing results of Hegarty-Miller on phase
transitions. Finally, we find the smallest sizes of MSTD pairs.

21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th