  Many problems in additive number theory, such as Fermat's last theorem and
the twin prime conjecture, can be understood by examining sums or differences
of a set with itself. A finite set $A \subset \mathbb{Z}$ is considered
sum-dominant if $|A+A|>|A-A|$. If we consider all subsets of ${0, 1, ...,
n-1}$, as $n\to\infty$ it is natural to expect that almost all subsets should
be difference-dominant, as addition is commutative but subtraction is not;
however, Martin and O'Bryant in 2007 proved that a positive percentage are
sum-dominant as $n\to\infty$.
  This motivates the study of "coordinate sum dominance". Given $V \subset
(\Z/n\Z)^2$, we call $S:={x+y: (x,y) \in V}$ a coordinate sumset and $D:=\{x-y:
(x,y) \in V\}$ a coordinate difference set, and we say $V$ is coordinate sum
dominant if $|S|>|D|$. An arithmetically interesting choice of $V$ is
$\bar{H}_2(a;n)$, which is the reduction modulo $n$ of the modular hyperbola
$H_2(a;n) := {(x,y): xy \equiv a \bmod n, 1 \le x,y < n}$. In 2009, Eichhorn,
Khan, Stein, and Yankov determined the sizes of $S$ and $D$ for
$V=\bar{H}_2(1;n)$ and investigated conditions for coordinate sum dominance. We
extend their results to reduced $d$-dimensional modular hyperbolas
$\bar{H}_d(a;n)$ with $a$ coprime to $n$.
