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 $\mathbb{F}$, in particular the special case $\mathbb{F}={\mathbb{F}}_p$. They are unified by the theme of "breaking the $3/2$ threshold", epitomising the previous state of the art. These estimates stem from specially suited applications of incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. We establish the estimate $|R[A]|\gtrsim |A{|}^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios defined by triples of elements of a (sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for the rest of the estimates) set $A\subset \mathbb{F}$, pinned at infinity. The cross-ratio naturally arises in various sum-product type questions of projective nature and is the unifying concept underlying most of our results. It enables one to take advantage of its symmetry properties as an onset of growth of, for instance, products of difference sets. The geometric nature of the cross-ratio enables us to break the version of the above threshold for the minimum number of distinct triangle areas $Ou{u}^{\prime }$, defined by points $u,u^{\prime }$ of a non-collinear point set $P\subset {\mathbb{F}}^2$. Another instance of breaking the threshold is showing that if $A$ is sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim M^{-2}|A{|}^{14/9}$. This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set $A\times A\subset {\mathbb{F}}^2$, the quantity often arising in applications of geometric incidence estimates.

59 pages. Added Theorem 2 and improved Theorem 1. This version is different to the one in print

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Cited by in corpus (19)

• On the restriction problem for discrete paraboloid in lower dimension
• Some remarks on the asymmetric sum--product phenomenon
• On the directions determined by Cartesian products and the clique number of generalized Paley graphs
• On $k$-point configuration sets with nonempty interior
• On distinct cross-ratios and related growth problems
• Collinear triples and quadruples for Cartesian products in ${\mathbb{F}}_p^2$
• Products of Differences over Arbitrary Finite Fields
• Bounds of Trilinear and Trinomial Exponential Sums
• Exact values and improved bounds on the clique number of cyclotomic graphs
• Additive decompositions of large multiplicative subgroups in finite fields
• Asymptotics for the number of directions determined by $[n]\times [n]$ in ${\mathbb{F}}_p^2$
• Incidences of Möbius transformations in ${\mathbb{F}}_p$
• A sum-bracket theorem for simple Lie algebras
• Szemerédi-Trotter type results in arbitrary finite fields
• Group actions, the Mattila integral and applications
• An effective local-global principle for algebraic varieties and the sum product problem in finite fields
• A short note on the multiplicative energy of the spectrum of a set
• Conditional expanding bounds for two-variable functions over arbitrary fields
• On growth of the number of determinants with restricted entries