We present a new method to bound the cardinality of triple product sets in groups and give three applications. A new and unexpectedly short proof of the Plunnecke-Ruzsa sumset inequalities for Abelian groups. A new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made. A further generalisation of the Plunnecke-Ruzsa inequalities in general groups.

New Proofs of Pl\"unnecke-type Estimates for Product Sets in Groups

Higher moments of convolutions

In this article we survey some of the recent developments in the structure theory of set addition.

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The structure theory of set addition revisited

We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension $${\delta}$$
 of the limit set close to $${{n-1\over 2}}$$
 . The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors–David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.

Spectral gaps, additive energy, and a fractal uncertainty principle

This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, as well as other fields. We discuss linear algebraic groups, with SL_2(Z/pZ) as the basic example, as well as permutation groups. The emphasis lies on the ideas behind the methods.

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Growth in groups: ideas and perspectives

We present a new method to bound the cardinality of product sets in groups and give three applications. A new and unexpectedly short proof of the Plunnecke-Ruzsa sumset inequalities for commutative groups. A new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made. A further generalisation of the Plunnecke-Ruzsa inequalities in general groups.

New proofs of Plünnecke-type estimates for product sets in groups

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 $Ouu'$, defined by points $u,u'$ 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.

New results on sum-product type growth over fields

We study the Erdos distinct distance conjecture in the plane over an arbitrary field $\mathbb{F}$, proving that any $A$ with $|A|\leq\mathrm{char}(\mathbb{F})^{4/3}$ in positive characteristic, either determines $\gg |A|^{2/3}$ distinct pairwise distances from some point of $A$ to its other points, or the set $A$ lies on an isotropic line. We also establish for the special case of the prime residue field $\mathbb{F}_p$ that the condition $|A|\geq p^{5/4}$ suffices for $A$ to determine a positive proportion of the feasible $p$ distances. This significantly improves prior results on the problem.

On the Pinned Distances Problem over Finite Fields

We prove that the clique number of the Paley graph is at most $\sqrt{p/2} + 1$, and that any supposed additive decompositions of the set of quadratic residues can only come from co-Sidon sets.

Giorgis Petridis

Papers

New Proofs of Pl\"unnecke-type Estimates for Product Sets in Groups

New proofs of Plünnecke-type estimates for product sets in groups

New results on sum-product type growth over fields

On the Pinned Distances Problem over Finite Fields

Refined estimates concerning sumsets contained in the roots of unity