Coordinate sum and difference sets of $d$-dimensional modular hyperbolas

Abstract

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 $V\subset (\mathbb{Z}/n\mathbb{Z}{)}^2$, 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 $H_2(a;n):=(x,y):xy\equiv a\mathrm{mod}n,1\le x,y<n$. 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 $n$.

Version 1.0, 14 pages, 2 figures