Some remarks on sets with small quotient set

TL;DR

This paper investigates the structure of finite real number sets with small quotient sets, establishing lower bounds on difference and multiple sets sizes, advancing understanding in additive combinatorics.

Contribution

It proves new lower bounds on difference and tripling set sizes for sets with small quotient sets, extending previous results in additive combinatorics.

Findings

|A-A| > |A|^{5/3 - o(1)} for sets with small quotient set

|3A| > |A|^{2 - o(1)} for such sets

Provides bounds that improve understanding of set growth in additive number theory

Abstract

We prove, in particular, that for any finite set of real numbers A with |A/A| \ll |A| one has |A-A| > |A|^{5/3 - o(1)}. Also we show that |3A| > |A|^{2-o(1)} in the case.