On Sets with More Products than Quotients

Hung Viet Chu

Published 2019-07-31Version 1

Given a finite set $A\subset \mathbb{R}\backslash \{0\}$, define \begin{align*}&A\cdot A \ =\ \{a_i\cdot a_j\,|\, a_i,a_j\in A\},\\ &A/A \ =\ \{a_i/a_j\,|\,a_i,a_j\in A\}, &A + A \ =\ \{a_i + a_j\,|\, a_i,a_j\in A\},\\ &A - A \ =\ \{a_i - a_j\,|\,a_i,a_j\in A\}.\end{align*} 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 $|A + A|>|A - A|$. Though much research has been done on MSTD sets, research on MPTQ sets hardly grows at the same pace. While many properties of MSTD sets still hold for MPTQ sets, MPTQ sets have many unique properties. This paper examines the search for MPTQ sets, when sets are not MPTQ, and what sequences do not contain MPTQ subsets.