ON THE MINIMUM CARDINALITY OF MPTQ SETS

Abstract

Given a finite set A ⊂ R, we define A + A = {a + a′ | a, a′ ∈ A} and A − A = {a − a′ | a, a′ ∈ A}. A set A is said to be an MSTD (More Sums than Differences) set if |A+A| > |A−A|. We define A.A = {aa′ | a, a′ ∈ A} and A/A = {a/a′ | a, a′ ∈ A, a′ ≠ 0}. Analogous to MSTD sets, H V Chu defines a set A ⊂ R ∖ {0} to be an MPTQ (More Products than Quotients) set if |A.A| > |A/A|. It is known by the exponentiation of MSTD sets that there exist MPTQ sets of cardinality 8. In an attempt to determine the smallest cardinality of an MPTQ set, Chu proved that an MPTQ set of real numbers must have at least 5 elements. In this work, we prove that a set of real numbers with cardinality 5 is not an MPTQ set. So we conclude an MPTQ set of real numbers must contain at least 6 elements. We have identified certain cases of sets with cardinality 6 that are not MPTQ sets. Further, we give an infinite family of MPTQ sets that are not the exponential of an MSTD set. © 2024, Colgate University. All rights reserved.