# On Sets with More Products than Quotients

## TL;DR

This paper investigates sets with more products than quotients, showing their rarity among finite subsets of real numbers, establishing minimal sizes for such sets, and analyzing specific sequences lacking these sets.

## Contribution

It introduces an efficient method to find MPTQ sets, proves minimal size bounds, and explores sequences that do not contain MPTQ subsets.

## Key findings

- Proportion of MPTQ sets approaches zero as n increases.
- Minimum size of MPTQ sets of positive numbers is 8.
- Minimum size of MPTQ sets with both positive and negative numbers is 5.

## Abstract

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 $|A + A|>|A - A|$. 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 $\{1,2,\ldots,n\}$ and proves that as $n\rightarrow \infty$, 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.