Constructions of generalized MSTD sets in higher dimensions

Fringe Pairs in Generalized Mstd Sets

A More Sums Than Differences (MSTD) set is a set A for which |A+A| > |A−A|. Martin and O’Bryant proved that the proportion of MSTD sets in {0, 1, . . . , n} is bounded below by a positive number as n goes to infinity. Iyer, Lazarev, Miller and Zhang introduced the notion of a generalized MSTD set, a set A for which |sA− dA| > |σA − δA| for a prescribed s + d = σ + δ. We offer efficient construc...

Explicit Constructions of Infinite Families of Mstd Sets

We explicitly construct infinite families of MSTD (more sums than differences) sets, i.e., sets where |A+A| > |A−A|. There are enough of these sets to prove that there exists a constant C such that at least C/r of the 2 subsets of {1, . . . , r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/2 for some polynomial f(r)). We co...

Constructions of generalized Sidon sets

We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k = s1 + s2, si ∈ S; such sets are called Sidon sets if g = 2 and generalized Sidon sets (or B2[ ⌈ g/2 ⌉ ] sets) if g ≥ 3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Koulantzakis’ idea of interleaving sever...

Two Generalized Constructions of Relative Difference Sets

We give two generalizations of some known constructions of relative difference sets. The first one is a generalization of a construction of RDS by Chen, Ray-Chaudhuri and Xiang using the Galois ring GR(4,m). The second one generalizes a construction of RDS by Ma and Schmidt from the setting of chain rings to a setting of more general rings.

Symmetries of Julia sets in higher dimensions