New Results on Sum‐product Type Growth Over Fields

Abstract

We prove a range of new sum-product type growth estimates over a general field F, in particular the special case F = F p . They are unified by the theme of "breaking the 3/2 threshold", epitomising the previous state of the art.This concerns two pivotal for the sum-product theory questions, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of F, as well as the number of values of the symplectic form determined by a finite subset of F 2 .We establish the estimate |R[A]| |A| 8/5 for cardinality of the set R[A] of distinct cross-ratios, defined by triples of elements of a (sufficiently small if F has positive characteristic, similarly for the rest of the estimates) set A ⊂ F, pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set P ⊂ F 2 , the number of distinct values of the symplectic form ω on pairs of distinct points u, u ′ of P is |ω(P )| |P | 2/3+c , with an explicit c. Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein.Our proofs strongly use specially suited applications of new incidence bounds over F, which apply to higher moments of representation functions. The technical thrust of the paper is using additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any F, by showing that if A is sufficiently small and has additive doubling constant M , then * |AA| M −2 |A| 14/9 . This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set A× A ⊂ F 2 , the quantity often arising in applications of geometric incidence estimates. (Expansion for large sets) IfF = F p , then |R[A]| ≫ min p, |A| 5/2 p 1/2 . In particular, if |A| ≥ p 3/5 , then |R[A]| ≫ p. 3. (Uniform expansion) If F = F p , then |R[A]| ≫ min{p, |A| 3 2 + 1 22 log − 4 9 |A|}.

Stronger sum-product inequalities for small sets

“…Theorem 3 enables one to improve upon (4). This can also be viewed as the special case of the general open question, concerning the minimum cardinality of set of values of the symplectic form on pairs of points in a given set in the plane F 2 (here the set being A × A), see [MPORS,Theorem 4] for a general geometric bound. We formulate the next theorem in slightly more generality.…”

11. Point-plane incidences and some applications in positive characteristic

“…For applications over the prime residue field F p there is the following asymptotic version. See [33,Theorem 8] and [32,Section 3] for its (easy) derivation from Theorem 1.…”

“…Theorem 1 has recently found many applications in sum-product type estimates in, e.g., [38], [3], [33] where the arising sets of points and planes have natural structure of Cartesian products. In particular, in [3,Corollary 6], it was observed that Theorem 1 implied a pointline incidence bound in F 2 in the special case of the point set being a Cartesian product.…”

“…Now, in the application of Theorem 1B with each fixed z, we forbid the set of lines arising at infinity after projecting the conics in L ′ , containing z. For this restricted number of incidences one has k = k 0 in estimate (33).…”

A New Sum–product Estimate in Prime Fields

“…We note that their condition for |A| < p 3/5 can be extended to |A| < p 2/3 , see Remark 10 for more details. See also [12] for variations on the sum-product theorem including sharper results for the few sums many products problem, see [13] for the few products many sums problem and [11] for various other results related to expanders in prime fields.…”