!newop ACF
!newop eq
!newop Zar
!newop st
!newop stp
!newop loc
!newop cl
!newop trd
!newop dim
!newop inc
!newop len
!newop locus
!newop Stab
!newop ccl
!newop tp
!newop dcl
!newop acl
!newop Cb
!newop Aut
!newop End
!newop Mat
!newop Isog
!newcal U
!newcal G
!newbb R
!newbb C
!newbb Z
!newbb Q
!newbb N

\renewcommand{\P}{\mathcal{P}}

\providecommand{\stR}{{ }^*\R}
\providecommand{\stC}{{ }^*\C}
\providecommand{\dl}{\delta}
\providecommand{\bdl}{\boldsymbol\dl}
\providecommand{\fdl}{\dl_{\"fin}}
\providecommand{\clbdl}{\cl_{\bdl}}
\providecommand{\dind}{\|/^{\bdl}}
\providecommand{\ndind}{\+/^{\bdl}}
\providecommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\providecommand{\calE}{\mathcal{E}}
\renewcommand{\a}{\overline{a}}
\renewcommand{\b}{\overline{b}}
\renewcommand{\c}{\overline{c}}
\renewcommand{\d}{\overline{d}}
\newcommand{\g}{\overline{g}}
\newcommand{\h}{\overline{h}}
\newcommand{\x}{\overline{x}}
\newcommand{\y}{\overline{y}}
\newcommand{\z}{\overline{z}}
\newcommand{\gammatup}{\overline{\gamma}}
\newcommand{\zetatup}{\overline{\zeta}}
\newcommand{\pp}{\mathfrak{p}}
\newcommand{\qq}{\mathfrak{q}}
\renewcommand{\O}{\mathcal{O}}
\providecommand{\AA}{\mathbb{A}}
\renewcommand{\L}{\mathcal{L}}
\renewcommand{\P}{\mathbb{P}}

\newcommand{\HQ}{\mathcal{H}_\Q}
\newcommand{\HZ}{\mathcal{H}_\Z}

\newcommand{\Lint}{\mathcal{L}_{\"int}}
%\newcommand{\Km}{ {{ }^*K} }
\newcommand{\Km}{ \mathbb{K} }
%\newcommand{\Km}{ \mathcal{U} }
\newcommand{\Kmeq}{\Km^{\operatorname{eq0}}}

\providecommand{\dcleq}{\dcl^{\eq}}
\providecommand{\acleq}{\acl^{\eq}}

\newcommand{\defnstyle}[1]{\textbf{#1}}

\newcommand{\zarcl}[1]{\overline{#1}^{\Zar}}

\newcommand{\interalg}{~~^0}

\newcommand{\modsim}{/\negmedspace\sim}

\providecommand{\concat}{\^}

\newcommand{\Geom}{\mathcal{G}}
\newcommand{\KGeom}{\Geom_K}

\newcommand{\Keq}{K^{\eq}}

\newcommand{\disjointunion}{\mathbin{\dot{\cup}}}
\newcommand{\Disjointunion}{\mathbin{\dot{\bigcup}}}

\newcommand{\sideremark}[1]{{\it \small (#1)}}

\theoremstyle{remark}
\newtheorem{setup}[theorem]{Setup}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{recall}[theorem]{Recall}
\newtheorem{answer}[theorem]{Answer}

% Rahim's nonforking symbol (I don't know where got it from)
\def\Ind#1#2{#1\setbox0=\hbox{$#1x$}\kern\wd0\hbox to 0pt{\hss$#1\mid$\hss}
\lower.9\ht0\hbox to 0pt{\hss$#1\smile$\hss}\kern\wd0}
\def\ind{\mathop{\mathpalette\Ind{}}}
\def\notind#1#2{#1\setbox0=\hbox{$#1x$}\kern\wd0\hbox to 0pt{\mathchardef
\nn=12854\hss$#1\nn$\kern1.4\wd0\hss}\hbox to
0pt{\hss$#1\mid$\hss}\lower.9\ht0 \hbox to
0pt{\hss$#1\smile$\hss}\kern\wd0}
\def\nind{\mathop{\mathpalette\notind{}}}


## Setup
# Pseudofinite dimension

***)
  $\U (= \P(\omega)$ non-principal ultrafilter.
**
  $K := \C^\U$.
**
  $X (= K^n$ is \defnstyle{internal} if $X = \prod_{s-->\U}X_s$ for some 
  $X_s (= \C^n$.

**
  Then set $|X| := \prod_{s-->\U} |X_s| (- \N^\U \cup \{\infty\}$.

** Fix $\xi (- \N^\U$ with $\xi > \N$.

Definition[Coarse pseudofinite dimension $\bdl$]:
  For $X$ internal,

  \[ \bdl(X) = \bdl_{\xi}(X) := \st ({ \frac{\log(|X|)}{\log(\xi)} }) (- 
  \R_{>=0} \cup \{-\infty,\infty\} .\]

  .
%** $\bdl(X) < c (- \R_{>0}$ iff for some $\epsilon>0$ and $A (- \U$,
%$|X_s| <= O((\xi_{0,s})^{c-\epsilon})$ for $s (- A$.
%** $\bdl(X\times Y)=\bdl(X)+\bdl(Y)$.
%** $\bdl(X \cup Y) = \max(\bdl(X),\bdl(Y))$.

** {\em Note that internality is closed under cardinality quantifiers:}
  If $R (= K^n \times K^m$ is internal and $\alpha (- \R^\U$,
  then $\{ \y (- K^n : \exists_{>=\alpha} \x. R(\x,\y) \}$ is internal.
*e)*



# $\Lint$ monster
***) $\Lint$: predicate for each internal $X (= K^n$.
** $\Km ~> K$ monster model in $\Lint$.
** For $\phi (- \Lint$, set $\bdl(\phi) := \bdl(\phi(K))$.
** $\bdl$ has a unique extension to $(\Lint)_{\Km}$ such that
  *\[a*
    \tp(\b) &|-> \bdl(\phi(\x,\b)) \\\
    S_{\y}({}) &--> \{-\infty\} \cup \R \cup \{\infty\}
  *a\]*
  is well-defined and continuous for each $\phi(\x,\y) (- \Lint$.
** Explicitly, $\bdl(\phi(\x,\a)) := \sup \{ q (- \Q : \Km |= 
\exists_{>=\xi^q} \x.\; \phi(\x,\a) \}$.
** For $\Phi$ a partial type, $\bdl(\Phi) := \inf \{ \bdl(\phi) : \Phi |= 
\phi \}$.
** $\bdl(a/C) := \bdl(\tp(a/C))$.

Fact:
  For $C (= \Km$ small and $a,b (- \Km^{<\omega}$,
  **(i) $a==_C b ==> \bdl(a/C)=\bdl(b/C)$.
  **(ii) $\bdl(ab/C)=\bdl(a/bC)+\bdl(b/C)$.
  **(iii) A partial type $\Phi$ over $C$ has a realisation
  $a (- \Phi(\Km)$ with $\bdl(a/C)=\bdl(\Phi)$.
  *ee*
  .
*e)*



# $\acl^0$

We have $\C <= \C^{\U} <= \Km$.

Definition:
  {\em Superscript 0 means: reduct to $\"ACF_\C$.}

  Work in $\Kmeq := \{ \"ACF-\textrm{imaginaries} \}$ (``$= \Km^{<\omega}$ '').

  ***) $d^0(B) := \trd(B/\C)$
  ** $a (- \acl^0(B)$ iff $d^0(a/B) = \trd(a/\C(B)) = 0$.
  %** $\Cb^0(a/B) := \Cb^{ACF}(a/\C(B))$
  *e)*
  .
Remark:
  $a (- \acl^0(B) ==> \bdl(a/B) = 0$.
  .



## Coherence
% Stability theory teaches us that given a dimension function, we should 
% look for pregeometries. Can't hope to say much about \bdl in general, but 
% here's a case where we can.

Definition:
  $P (= \Km$ is \defnstyle{coherent} if
  for any tuple $\a (- P^{<\omega}$,
  \[ \bdl(\a) = d^0(\a) .\]
  .

{\em In other words, $\bdl$ is equal on $P^{<\omega}$ to the dimension function 
of the pregeometry $(P;\acl^0)$.}


{\em More generally:}

Definition:
  $a (- \Kmeq$ is in \defnstyle{coarse general position} (or is 
  \defnstyle{cgp}) if for any $B (= \Km$,
  \[ d^0(a/B) < d^0(a) ==> \bdl(a/B) = 0 .\]
  .

Any $a (- \Km$ is cgp.

Definition:
  $P (= \Kmeq$ is \defnstyle{coherent} if
  ***) every $a (- P$ is cgp, and
  ** for any tuple $\a (- P^{<\omega}$,
  \[ d^0(\a)=\bdl(\a) .\]
  *e)*
  .

Then 
***) $(P;\acl^0)$ is a pregeometry,
** if $d^0(a)=k \; (\forall a (- P)$,
  then $\bdl(\a) = k\dim_P(\a)$ for any $\a (- P^{<\infty}$.
*e)*


# Modularity of coherence

Definition:
  For $P (= \Kmeq$,
  \[ \ccl(P) := \{ x (- \acl^0(P) : \{x\} \;\textrm{is coherent}\} .\]
  .

Lemma:
  If $P$ is coherent, so is $\ccl(P)$.
  .

Proposition:
  Suppose $P = \ccl(P)$ is coherent.

  Then $(P;\acl^0)$ is a modular pregeometry.

  (i.e. for $a,b (- P$ and $C (= P$,
  if $a (- \acl^0(bC) \\ (\acl^0(C) \cup \acl^0(b))$
  then exists $c (- \acl^0(C)$ such that $a (- \acl^0(bc)$.)
  .
{\em (Eventual) consequence of Szemeredi-Trotter theorems}

Example:
  Suppose $G$ is a complex abelian algebraic group,
  $F <= \End^0(G) := \Q (x)_\Z \End(G)$ a division ring,
  $A (= G(\Km)$ independent generics.

  Let $Q := \{ \Sigma_i \eta_i(a_i) : \eta_i (- F \cap \End(G); a_i (- A \} (= \Kmeq$.

  Then $(Q;\acl^0)$ is a modular pregeometry.
  .

Theorem:
  Up to $\acl^0$-interalgebraicity,
  any coherent $P (= \Kmeq$ is contained in a union of finitely many 
  orthogonal such $Q$.
  .
Proof[Proof ingredients]:
  Veblen-Young co-ordinatisation, plus generalisation of Evans-Hrushovski.
  .

## Elekes-Szabó consequences
Definition:
  Say a finite subset $X$ of a variety $W$ is \defnstyle{$\tau$-cgp} if for 
  any proper subvariety $W' (!= W$ of complexity $<=\tau$, we have
  $|X \cap W'| < |X|^{\frac1\tau}$ .
  .

Definition:
  If $V (= \prod_i W_i$ are irreducible complex algebraic varieties,
  with $\dim(W_i) = m$ and $\dim(V) = d m$,
  say $V$ \defnstyle{admits a powersaving} if for some $\tau$ and 
  $\epsilon>0$
  \[ |{ \prod_i X_i \cap V }| <= O(N^{d-\epsilon}) \]
  for $\tau$-cgp $X_i (= W_i$ with $|X_i| <= N$.
  .
Lemma:
  $V$ admits no powersaving iff exists coherent generic $\a (- V(\Km)$.
  .


Definition:
  $H <= G^n$ is a \defnstyle{special subgroup} if $G$ is a commutative 
  complex algebraic group and $H = \ker(A)^o$ for some $A (- \"Mat(F \cap \End(G))$ 
  for some division subalgebra $F <= \End^0(G) := \Q (x)_{\Z} \End(G)$.
  .
Example:
  $G = \C^n$; then $\End^0(G) = \End(G) = \Mat_n(\C)$,
  and $H <= G^l$ is special iff it is a subspace defined by $F$-linear equations
  for some division subalgebra $F <= \Mat_n(\C)$.
  .

Theorem:
  $V (= \prod_i W_i$ admits no powersaving iff it is in co-ordinatewise 
  algebraic correspondence with a product of special subgroups.
  .

Where
Definition:
  Co-ordinatewise algebraic correspondence blah blah.
  .


## Applications
# Generalised sum-product phenomenon

Corollary:
  Let $(G_1,+_1)$ and $(G_2,+_2)$ be one-dimensional non-isogenous connected 
  complex algebraic groups,
  and for $i=1,2$ let $f_i : G_i(\C) \rightarrow  \C$ be a rational map.
  Then there are $\epsilon,c>0$ such that if $A \subset \C$ is a finite set 
  lying in the range of each $f_i$, then setting $A_i=f_i^{-1}(A) \subseteq 
  G_i(\C)$ we have
  \[ \max( |A_1 +_1 A_1|, |A_2 +_2 A_2| ) \geq  c|A|^{1+\epsilon} .\]
  .
Proof:
  Else,
  \[ \{ (g_1,h_1,g_1 +_1 h_1, g_2, h_2, g_2 +_2 h_2) : f_1(g_1) = f_2(g_2), f_1(h_1) = f_2(h_2) \} \]
  admits no powersaving,
  so get group $(G;+)$ such that $\Gamma_{+_i}$ is in co-ordinatewise 
  correspondence with $\Gamma_+$, $i=1,2$.
  
  But then (by Ziegler) $G_i$ is isogenous to $G$.
  .

{\em Similarly in higher dimension, with a cgp assumption.}



# Intersections of varieties with approximate subgroups
Theorem:
  $\Gamma \leq G(\Km)$ a ${}$-$\bigwedge$-definable subgroup of a 
  1-dimensional algebraic group $G$, with $\bdl(\Gamma) = \dim(G)$.

  Then any coherent tuple $\gammatup \in \Gamma^n$ is generic in a coset of 
  an algebraic subgroup of $G^n$.
  .
Similarly in higher dimension, with a cgp assumption.


Corollary:
  Let $G$ be a commutative complex algebraic group.
  Suppose $V$ is a subvariety of $G^n$ which is not a coset of a subgroup.
  Then there are $N, \epsilon, \eta>0$ depending only on $G$ and the 
  complexity of $V$ such that if $A \subseteq  G$ is a finite subset such 
  that $A-A$ is $\tau$-$cgp$ and $|A+A| \leq  |A|^{1+\epsilon}$ and $|A| 
  \geq  N$, then
  $|A^n \cap V| < |A|^{\frac{\dim(V)}{\dim(G)} - \eta}$.
  .



# Diophantine connection
%  Corollary:
%    Let $G$ be an algebraic group over $\C$.
%    Let $\Gamma <= G(\Km)$ be a ${}$-definable cgp subgroup,
%    and $\bdl(\Gamma)=\dim G$.
%
%    Suppose $\xitup (- \Gamma^n$ is coherent.
%    Then $\locus^{G^n}(\xitup/\C)$ is a coset of an algebraic subgroup.
%    .

Example:
  $G=E$ complex elliptic curve.

  $E[\infty] := \bigcup_m E[m]$ torsion subgroup.

  Suppose $V (= E^n$ is an irreducible closed complex subvariety such that 
  $V(\C) \cap E[\infty]$ is Zariski dense in $V$. Let $d := \dim(V)$.

  By Manin-Mumford, $V$ is a coset of an algebraic subgroup.
  Hence for any $\epsilon>0$, for arbitrarily large $r (- \N$,
  \[ |V(\C) \cap E[r!]^n| >= |{ E[r!] }|^{d-\epsilon} .\]

  Suppose conversely that we only know this consequence of Manin-Mumford on 
  the asymptotics of the number of torsion points in $V$. Then $V$ has a 
  coherent generic non-standard torsion point, and so by above theorem $V$ 
  is a coset.

  Similarly for Mordell-Lang.
  .



## Sharpness
Consider case $G=\G_a^n$, so $\End^0(G) = \End(G) = \Mat_n(\C)$.

Want to show: if $F <= \Mat_n(\C)$ is a division ring,
and $H <= G^l$ is a subspace defined by $F$-linear equations,
then exists $\zetatup (- H$ coherent generic.

Fact[Amitsur-Kaplansky]:
  Any division subring $F (= \Mat_n(\C)$ has finite dimension over its centre.
  .
Corollary:
  Exists finitely generated subring $\O (= F$ such that $H$ is defined by 
  linear equations with coefficients from $\O$ and $\O$ is 
  \defnstyle{constrainedly filtered}: there are finite $\O_n (= \O$ such that
  \begin{enumerate}[(CF0)]
  **(CF0) $\O_n (= \O_{n+1}$; $\bigcup_{n (- \N} \O_n = \O$
  **[(CF1)] $\exists k.\; \forall n.\; \O_n+\O_n (= \O_{n+k}$;
  **[(CF2)] $\forall a (- \O.\; \exists k.\; \forall n.\; a\O_n (= 
  \O_{n+k}$;
  **[(CF3)] $\forall \epsilon>0.\; \frac{|\O_{n+1}|}{|\O_n|} <= 
  O(|\O_n|^\epsilon)$.
  *ee*
  .

(e.g. $\Z = \bigcup_n [-2^n,2^n]$ is constrainedly filtered.)

Let ``$X_k := \prod_{s --> \U}(\sum_{i=1}^s \O_{s-k}\gamma_i)$'' with 
$\gamma_i (- G$ generic independent.

Then $X := \bigcap_k X_k$ is an $\O$-submodule and $\bdl(X)=\bdl(X_0)$,
and $\zetatup (- H \cap X^l$ with $\bdl(\zetatup) = \bdl(H \cap X^l)$ is 
coherent.


{\em General $G$ similar; always $\End^0(G)$ embeds in $\Mat^{\dim(G)}(\C)$, 
by Lie theory.}



## Relaxing general position

Remark:
  $V :=$ graph of $(a_1,b_1)*(a_2,b_2) = (a_1+a_2+b_1^2b_2^2,b_1+b_2)$,
  $X_i := \{-N^4,...,N^4\}\times\{-N,...,N\} (= \C^2 =: W_i$.

  Then $|X_i^3 \cap V| >= \Omega(|X_i|^2)$, but not in coarse general 
  position, and $V$ is not in co-ordinatewise correspondence with the graph 
  of a group operation.
  .

