Stronger sum-product inequalities for small sets

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 inequalitydue to Roche-Newton, Rudnev and Shkredov, toas well as |AA| 36 |A − A| 24 ≫ |A| 73−o(1) .The latter inequality is "threshold-breaking", for it shows for ǫ > 0, one has |AA| |A| 1+ǫ ⇒ |A − A| ≫ |A| 3 2 +c(ǫ) , with c(ǫ) > 0 if ǫ is sufficiently small. This implies that regardless of ǫ, |AA − AA| ≫ |A| 3 2 + 1 56 −o(1) .

On products of shifts in arbitrary fields

“…The 6/5 threshold has recently been broken, see [1], [2], and [3]. The following theorem was proved in [2] by Rudnev, Shakan, and Shkredov, and is the current state of the art bound.…”

New bounds for distance-type problems over prime fields

“…For any w ∈ A 2 − A 2 , let P w := (A 2 − A 2 ) ∩ (P − w). One can follow the first paragraph of the proof of [14,Theorem 3] to prove the following lemma.…”

“…is a solution of (13). In other words, there are at least |A| 4 tuples (a, b, c, d) ∈ A 4 which gives us solutions of (13) in the form of (14).…”

“…To prove Theorem 1.5, we use a new sum-product idea which has been introduced recently by Rudnev, Shakan, and Shkredov [14,Theorem 3]. Note that in [14] Rudnev et al proved that for sufficiently small A in F p , if |AA| ≤ |A| 1+ǫ for some positive small ǫ, then |A − A| ≥ |A| 3 2 +c(ǫ) . As a consequence of this result, they derived that |AA−AA| ≫ |A| 3 2 + 1 56 .…”

“…is still an open question. We refer the interested reader to [14] for more discussions. To prove Theorem 1.1, we will make use of the following lemmas.…”