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April 2020 On sets with more products than quotients
Hùng Việt Chu
Rocky Mountain J. Math. 50(2): 499-512 (April 2020). DOI: 10.1216/rmj.2020.50.499

Abstract

Given a finite set $A\subset ℝ\setminus \left\{0\right\}$, define

$\begin{array}{cccc}\hfill A\cdot A& =\left\{{a}_{i}\cdot {a}_{j}\mid {a}_{i},{a}_{j}\in A\right\},\hfill & \hfill \phantom{\rule{1em}{0ex}}A∕A& =\left\{{a}_{i}∕{a}_{j}\mid {a}_{i},{a}_{j}\in A\right\},\hfill \\ \hfill A+A& =\left\{{a}_{i}+{a}_{j}\mid {a}_{i},{a}_{j}\in A\right\},\hfill & \hfill \phantom{\rule{1em}{0ex}}A-A& =\left\{{a}_{i}-{a}_{j}\mid {a}_{i},{a}_{j}\in A\right\}.\hfill \end{array}$

The set $A$ is said to be MPTQ (more product than quotient) if $|A\cdot A|>|A∕A|$ and MSTD (more sum than difference) if $|A+A|>|A-A|$. Since multiplication and addition are commutative while division and subtraction are not, it is natural to think that MPTQ and MSTD sets are very rare. However, they do exist. This paper first shows an efficient search for MPTQ subsets of $\left\{1,2,\dots ,n\right\}$ and proves that as $n\to \infty$, the proportion of MPTQ subsets approaches $0$. Next, we prove that MPTQ sets of positive numbers must have at least $8$ elements, while MPTQ sets of both negative and positive numbers must have at least $5$ elements. Finally, we investigate several sequences that do not have MPTQ subsets.

Citation

Hùng Việt Chu. "On sets with more products than quotients." Rocky Mountain J. Math. 50 (2) 499 - 512, April 2020. https://doi.org/10.1216/rmj.2020.50.499

Information

Received: 15 August 2019; Revised: 20 September 2019; Accepted: 25 September 2019; Published: April 2020
First available in Project Euclid: 29 May 2020

zbMATH: 07210974
MathSciNet: MR4104389
Digital Object Identifier: 10.1216/rmj.2020.50.499

Subjects:
Primary: 11N99