WHEN SETS CAN OR CANNOT BE PRODUCT-DOMINANT
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Given a finite set π΄ β β, we define π΄ + π΄ = {a + aβ² β£ a, aβ² β π΄}, π΄ β π΄ = {a β aβ² β£ a, aβ² β π΄}, π΄ β π΄ = {aaβ² β£ a, aβ² β π΄}, π΄ β π΄ = {a β aβ² β£ a, aβ² β π΄, aβ² β 0}. A set π΄ is said to be sum-dominant or MSTD (More Sums than Differences) if |π΄ + π΄| > |π΄ β π΄|, and a set π΄ β β β {0} is said to be product-dominant or MPTQ (More Products than Quotients) if |π΄ β π΄| > |π΄ β π΄|. In this paper, we shall discuss several properties of MPTQ sets, investigate techniques of generating an infinite family of MPTQ sets, and identify some characterizations under which a finite set of numbers can or cannot be product-dominant. We confirm the existence of MPTQ sets of perfect squares and justify that πα΅Κ° powers of prime numbers do not contain any MPTQ set. We extend the notion of MPTQ sets to the multiplicative group β€β* and recognize their correspondence with the MSTD sets in β€βββ.
Abstract
Given a finite set π΄ β β, we define π΄ + π΄ = {a + aβ² β£ a, aβ² β π΄}, π΄ β π΄ = {a β aβ² β£ a, aβ² β π΄}, π΄ β π΄ = {aaβ² β£ a, aβ² β π΄}, π΄ β π΄ = {a β aβ² β£ a, aβ² β π΄, aβ² β 0}. A set π΄ is said to be sum-dominant or MSTD (More Sums than Differences) if |π΄ + π΄| > |π΄ β π΄|, and a set π΄ β β β {0} is said to be product-dominant or MPTQ (More Products than Quotients) if |π΄ β π΄| > |π΄ β π΄|. In this paper, we shall discuss several properties of MPTQ sets, investigate techniques of generating an infinite family of MPTQ sets, and identify some characterizations under which a finite set of numbers can or cannot be product-dominant. We confirm the existence of MPTQ sets of perfect squares and justify that πα΅Κ° powers of prime numbers do not contain any MPTQ set. We extend the notion of MPTQ sets to the multiplicative group β€β* and recognize their correspondence with the MSTD sets in β€βββ.
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