COORDINATE SUM AND DIFFERENCE SETS OF d-DIMENSIONAL MODULAR HYPERBOLAS

摘要

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 ? Z is considered sum-dominant if |A + A| > |A ? A|. If we consider all subsets of {0, 1, . . . , n ? 1}, as n → ∞, 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 → ∞. This motivates the study of “coordinate sum dominance.” Given V ? (Z/nZ)2, we call S := {x+y : (x, y) ∈ V } a coordinate sumset and D := {x ? y : (x, y) ∈ V } a coordinate difference set, and we say V is coordinate sum dominant if |S| > |D|. An arithmetically interesting choice of V is Hˉ2(a; n), which is the reduction modulo n of the modular hyperbola H2(a; n) := {(x, y) : xy ≡ a mod n, 1 ≤ x, y < n}. In 2009, Eichhorn, Khan, Stein, and Yankov determined the sizes of S and D for V = Hˉ2(1; n) and investigated conditions for coordinate sum dominance. We extend their results to reduced d-dimensional modular hyperbolas Hˉd(a; n) with a coprime to n.