Difference sets are not multiplicatively closed

TL;DR

The paper proves that the difference set of a finite real number set has significantly larger product and quotient sets, showing it cannot be multiplicatively closed, with similar results in prime fields.

Contribution

It establishes that difference sets inherently have large product and quotient sets, demonstrating they are not multiplicatively closed, extending to prime fields.

Findings

Difference sets have large product and quotient sets.

Multiplicative subgroups smaller than p^{4/5-ε} cannot be difference sets.

Results hold in both real numbers and finite prime fields.

Abstract

We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. A similar result takes place in the prime field F_p for sufficiently small D. It gives, in particular, that multiplicative subgroups of size less than p^{4/5-\eps} cannot be represented in the form A-A for any A from F_p.