Stronger sum-product inequalities for small sets · arXivDesk1808.08465Aug 25, 2018v2: Improved sum-product bound to match difference-product bound
Stronger sum-product inequalities for small sets
Misha Rudnev, George Shakan, Ilya Shkredov
Abstract
Let F be a field and a finite A⊂F be sufficiently small in terms of the characteristic p of F if p>0
. We strengthen the "threshold" sum-product inequality
|AA|³ |A± A|² ≫ |A|⁶ , hence |AA|+|A+A|≫ |A|¹+(1/5),
due to Roche-Newton, Rudnev and Shkredov, to
|AA|⁵ |A± A|⁴ ≫ |A|¹1-o(1) , hence |AA|+|A± A|≫ |A|¹+(2/9)-o(1),
as well as
∣AA∣36∣A−A∣24≫∣A∣73−o(1). The latter inequality is "threshold-breaking", for it shows for
, one has
∣AA∣≤∣A∣1+ε⇒∣A−A∣≫∣A∣23+c(ε), with
if
is sufficiently small. This implies that regardless of
,
∣AA−AA∣≫∣A∣23+561−o(1).