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\begin{document}
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\section{Setup}
\subsection{Pseudofinite dimension}

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

\item
  Then set $|X| := \prod_{s\rightarrow \U} |X_s| \in \N^\U \cup \{\infty\}$.

\item Fix $\xi \in \N^\U$ with $\xi > \N$.

\begin{definition}[Coarse pseudofinite dimension $\bdl$]
  For $X$ internal,

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

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

\item {\em Note that internality is closed under cardinality quantifiers:}
  If $R \subseteq  K^n \times K^m$ is internal and $\alpha \in \R^\U$,
  then $\{ \y \in K^n : \exists_{\geq \alpha} \x. R(\x,\y) \}$ is internal.
\end{itemize}



\subsection{$\Lint$ monster}
\begin{itemize}\item $\Lint$: predicate for each internal $X \subseteq  K^n$.
\item $\Km \succ  K$ monster model in $\Lint$.
\item For $\phi \in \Lint$, set $\bdl(\phi) := \bdl(\phi(K))$.
\item $\bdl$ has a unique extension to $(\Lint)_{\Km}$ such that
  \begin{align*}
    \tp(\b) &\mapsto  \bdl(\phi(\x,\b)) \\
    S_{\y}(\emptyset ) &\rightarrow  \{-\infty\} \cup \R \cup \{\infty\}
  \end{align*}
  is well-defined and continuous for each $\phi(\x,\y) \in \Lint$.
\item Explicitly, $\bdl(\phi(\x,\a)) := \sup \{ q \in \Q : \Km \vDash  
\exists_{\geq \xi^q} \x.\; \phi(\x,\a) \}$.
\item For $\Phi$ a partial type, $\bdl(\Phi) := \inf \{ \bdl(\phi) : \Phi \vDash  
\phi \}$.
\item $\bdl(a/C) := \bdl(\tp(a/C))$.

\begin{fact}
  For $C \subseteq  \Km$ small and $a,b \in \Km^{<\omega}$,
  \begin{enumerate}[(i)]\item $a\equiv _C b \Rightarrow  \bdl(a/C)=\bdl(b/C)$.
  \item $\bdl(ab/C)=\bdl(a/bC)+\bdl(b/C)$.
  \item A partial type $\Phi$ over $C$ has a realisation
  $a \in \Phi(\Km)$ with $\bdl(a/C)=\bdl(\Phi)$.
  \end{enumerate}
\end{fact}
\end{itemize}



\subsection{$\acl^0$}

We have $\C \leq  \C^{\U} \leq  \Km$.

\begin{definition}
  {\em Superscript 0 means: reduct to $\operatorname{ACF}_\C$.}

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

  \begin{itemize}\item $d^0(B) := \trd(B/\C)$
  \item $a \in \acl^0(B)$ iff $d^0(a/B) = \trd(a/\C(B)) = 0$.
  %** $\Cb^0(a/B) := \Cb^{ACF}(a/\C(B))$
  \end{itemize}
\end{definition}
\begin{remark}
  $a \in \acl^0(B) \Rightarrow  \bdl(a/B) = 0$.
\end{remark}



\section{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.

\begin{definition}
  $P \subseteq  \Km$ is \defnstyle{coherent} if
  for any tuple $\a \in P^{<\omega}$,
  \[ \bdl(\a) = d^0(\a) .\]
\end{definition}

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


{\em More generally:}

\begin{definition}
  $a \in \Kmeq$ is in \defnstyle{coarse general position} (or is 
  \defnstyle{cgp}) if for any $B \subseteq  \Km$,
  \[ d^0(a/B) < d^0(a) \Rightarrow  \bdl(a/B) = 0 .\]
\end{definition}

Any $a \in \Km$ is cgp.

\begin{definition}
  $P \subseteq  \Kmeq$ is \defnstyle{coherent} if
  \begin{itemize}\item every $a \in P$ is cgp, and
  \item for any tuple $\a \in P^{<\omega}$,
  \[ d^0(\a)=\bdl(\a) .\]
  \end{itemize}
\end{definition}

Then 
\begin{itemize}\item $(P;\acl^0)$ is a pregeometry,
\item if $d^0(a)=k \; (\forall a \in P)$,
  then $\bdl(\a) = k\dim_P(\a)$ for any $\a \in P^{<\infty}$.
\end{itemize}


\subsection{Modularity of coherence}

\begin{definition}
  For $P \subseteq  \Kmeq$,
  \[ \ccl(P) := \{ x \in \acl^0(P) : \{x\} \;\textrm{is coherent}\} .\]
\end{definition}

\begin{lemma}
  If $P$ is coherent, so is $\ccl(P)$.
\end{lemma}

\begin{proposition}
  Suppose $P = \ccl(P)$ is coherent.

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

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

\begin{example}
  Suppose $G$ is a complex abelian algebraic group,
  $F \leq  \End^0(G) := \Q \otimes_\Z \End(G)$ a division ring,
  $A \subseteq  G(\Km)$ independent generics.

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

  Then $(Q;\acl^0)$ is a modular pregeometry.
\end{example}

\begin{theorem}
  Up to $\acl^0$-interalgebraicity,
  any coherent $P \subseteq  \Kmeq$ is contained in a union of finitely many 
  orthogonal such $Q$.
\end{theorem}
\begin{proof}[Proof ingredients]
  Veblen-Young co-ordinatisation, plus generalisation of Evans-Hrushovski.
\end{proof}

\section{Elekes-Szab\'o consequences}
\begin{definition}
  Say a finite subset $X$ of a variety $W$ is \defnstyle{$\tau$-cgp} if for 
  any proper subvariety $W' \subsetneq  W$ of complexity $\leq \tau$, we have
  $|X \cap W'| < |X|^{\frac1\tau}$ .
\end{definition}

\begin{definition}
  If $V \subseteq  \prod_i W_i$ 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 
  $\epsilon>0$
  \[ \left|{\prod_i X_i \cap V}\right| \leq  O(N^{d-\epsilon}) \]
  for $\tau$-cgp $X_i \subseteq  W_i$ with $|X_i| \leq  N$.
\end{definition}
\begin{lemma}
  $V$ admits no powersaving iff exists coherent generic $\a \in V(\Km)$.
\end{lemma}


\begin{definition}
  $H \leq  G^n$ is a \defnstyle{special subgroup} if $G$ is a commutative 
  complex algebraic group and $H = \ker(A)^o$ for some $A \in \operatorname{Mat}(F \cap \End(G))$ 
  for some division subalgebra $F \leq  \End^0(G) := \Q \otimes_{\Z} \End(G)$.
\end{definition}
\begin{example}
  $G = \C^n$; then $\End^0(G) = \End(G) = \Mat_n(\C)$,
  and $H \leq  G^l$ is special iff it is a subspace defined by $F$-linear equations
  for some division subalgebra $F \leq  \Mat_n(\C)$.
\end{example}

\begin{theorem}
  $V \subseteq  \prod_i W_i$ admits no powersaving iff it is in co-ordinatewise 
  algebraic correspondence with a product of special subgroups.
\end{theorem}

Where
\begin{definition}
  Co-ordinatewise algebraic correspondence blah blah.
\end{definition}


\section{Applications}
\subsection{Generalised sum-product phenomenon}

\begin{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
  \[ \max( |A_1 +_1 A_1|, |A_2 +_2 A_2| ) \geq  c|A|^{1+\epsilon} .\]
\end{corollary}
\begin{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 $G$.
\end{proof}

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



\subsection{Intersections of varieties with approximate subgroups}
\begin{theorem}
  $\Gamma \leq G(\Km)$ a $\emptyset $-$\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 $G^n$.
\end{theorem}
Similarly in higher dimension, with a cgp assumption.


\begin{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
  $|A^n \cap V| < |A|^{\frac{\dim(V)}{\dim(G)} - \eta}$.
\end{corollary}



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

\begin{example}
  $G=E$ complex elliptic curve.

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

  Suppose $V \subseteq  E^n$ 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 $r \in \N$,
  \[ |V(\C) \cap E[r!]^n| \geq  \left|{E[r!]}\right|^{d-\epsilon} .\]

  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.
\end{example}



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

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

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

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

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

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



\section{Relaxing general position}

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

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

\end{document}
