Recall motivation: CIT required for first-order axiomatisation of pseudo-exp,
specifically the Schanuel conjecture (over the kernel).

Let X (= G_m^N subvariety over \Qbar.

"Subgroup" of G_m^N means algebraic subgroup, i.e. of form 
    /\_i \Pi_j x_j^{n_ij} = 1
[and 'coset' means coset of connected such]

For H a subgroup (resp. coset) of G_m^N, a component S of H \cap X is a
_special subvariety_ (resp. _weakly special subvariety_) of X iff
    \dim S > \dim X - \codim H

Conjecture [CIT]:
    There are finitely many maximal special subvarieties.

Theorem [Weak CIT; Zilber / Bombieri-Masser-Zannier]:
    The union Z (= X of the dim>0 weakly special subvarieties of X is Zariski
    closed.
(Proof: Ax)
(Note: holds for X over \C)

Theorem [Habegger, BMZ]:
    There are only finitely special points on X\Z.
[Refinements due to Rémond ("over parameters" versions, at least for abelian
varieties), Maurin (X^oa |-> X^ta for X a curve)]

Now let C (= G_m^N be a curve / \Qbar.

Say N=3 [for concreteness].

X := C^2: note Z=X!

[n] : G_m^3 -> G_m^3; (x_1,x_2,x_3) |-> (x_1^n,x_2^n,x_3^n)

If x \in C and some [n]x \in C, with n >= 2, then call x an "echoing point" of
C, and [n]x its "echo". Then
    (x,[n]x)\in X \cap { (z_1,...,z_6) | z_4=z_1^n /\ z_5=z_2^n /\ z_6=z_3^n }
so (x,[n]x) is a special point of X.

Question: can C have infinitely many echoing points?

If N=2, then yes (consider dimensions).

If C is contained in a proper subgroup, C (= H < G_m^3, then again yes.

Else, CIT => no!

Remark: torsion points are echoing; MM => finitely many torsion points on C.


Let T (= G_m^3(\C) := { x  |  |x_1|=|x_2|=|x_3|=1 }.

Definition: evil locus := C(\C)\cap T

"Should" be finite: dim_R(C(\C)) = 2; dim_R(T) = 3; dim_R(G_m^3(C)) = 6...

but sometimes isn't...
    [Mingxi Wang: z |-> i(z+1)/(z-1)  is a bijection S^1 -> R\cup{\infty}]

Theorem [B-Habegger]:
    Suppose C (= G_m^N (N>=2, C / \Qbar) is contained in no proper subgroup
    and has finite evil locus. Then C has only finitely many echoing points.

Proof Part I : Ax and Pila-Wilkie 
---------------------------------

WMA C is _free_ - contained in no proper *coset*
    (if C is not free and has an echoing point, then C is contained in a
    proper subgroup).

Work in the o-minimal structure <\R;+,*,exp|> where exp| is *complex* exp
restricted to
    F := \R + i [0,2\pi)  (=  \C = \R+i\R ~= \R^2.
o-minimality is due to Wilkie.

Consider the definable family of definable sets
    E_n := { (k,x) \in \R^3 * F^3 | ( exp|(x) \in C /\ exp|(nx-k2i\pi) \in C }
    (so if k is integral, (k,x)\in E_n witnesses that exp|(x) is n-echoing)
and the projections to the k-coordinates
    K_n := { k | \exists x. (k,x) \in E_n }.

If exp|(x) is n-echoing, then (k,x) \in E_n for some k \in {0,1,...,n}^3.

Pila-Wilkie counting theorem:
    Let K_n^alg be the union of all positive-dimensional semi-algebraic
    subsets of K_n, and let K_n^tr := K_n \ K_n^alg.

    Then "rational points on K_n^tr are subpolynomially sparse":

    for all \epsilon > 0, exists c, for all n,B:
	| { q \in \Q^3 \cap K_n^tr  |  H(q) < B } | 
		    <=  c B^\epsilon
    where H(q) = max_i H(q_i),
    where H(a/b) = max{|a|,|b|}  (assuming (a,b)=1)
    (so for k\in\N, H(k) = k).

Claim: K_n^alg is empty for all n
Proof:
    Suppose S (= K_n^alg is a semi-algebraic curve; let, in an elementary
    extension, k\in S be generic over \R. So \trd(k/\R) = 1.
    Let R' := dcl(R,k) >= \R, let C' := R'^2 >= \C.

    Lemma [Ax]: 
      For x \in F(\Rfrak)^N
	  \trd(x,\exp|(x) / \C) > \ld_\Q(x / \C)
      unless x \in \C^N.
    Proof [after Wilkie]:
	Define derivation \delta(f(k)) := f'(k)
	    for f a definable function / \R;
	apply Ax's theorem.

    Now let \gamma := exp(k2i\pi)\in ([n]C - C).
    Then by Ax:
      \ld_\Q(k2i\pi / \C)  <=  \trd(k / \C) + \trd(\gamma / \C) - 1
			   <=	    1       +        2          - 1
			   <= 2 < N .
    
    C hence ([n]C - C) is free, so \gamma isn't generic, so
    \trd(\gamma / \C) = 1 hence \ld_\Q(k / \C) = 1.
    
    So say \pi : \G_m^N -->> \G_m^{N-1} such that \pi(\gamma) \in \C.
    By definition of K_n, say \alpha in C such that [n]\alpha-\gamma \in C.
    If \alpha \in \C then [n]\alpha-\gamma is generic in C but
    \pi([n]\alpha-\gamma)\in\C, contradicting freeness of C
    Hence \alpha is generic in C and \pi(\alpha) is generic in \pi(C).
    But then [n]\pi(\alpha)-\pi(\gamma)\in \pi(C) implies
    [n]\pi(C)-\pi(\gamma)=\pi(C), contradicting freeness of C.

Now for (n,k) integral, there are <\aleph_0 (k,x) \in E_n since there are
<\aleph_0 n-echoing points; so by uniform finiteness there is a
bound uniform in (n,k) on the number of such x. So Pila-Wilkie => for all
\epsilon > 0, exists c, for all n
    |{n-echoing points}| <= c n^\epsilon   (*)

Proof Part II: degree lower bounds
----------------------------------

Say C is over F.
Lemma: exists c such that for n large enough, if x is non-torsion and
    n-echoing, then
        \deg(x/F) := [F(x):F] >= c n^{1/11}.

But the conjugates of x over F are also n-echoing, so (picking \epsilon = 1/12
say) we contradict (*) if there are infinitely many non-torsion echoing
points.

By Manin-Mumford, there are only finitely many torsion points on C, so we're
done.

Sketch Proof of Lemma:
    Echoing points are algebraic.
        h : G_m^N -> \R
          ; (x_1,...,x_N) |-> \Sigma_i h(x_i)
    where
        h : \Qbar -> \R
          ; x |-> \Sigma_{v normalised abs value} max(0, log |x|_v)

    Fact [corollary of Rémond's toric Vojta inequality]:
        Echoing points have bounded height.
            [but not, a priori, their echos!]
    
    Now points of bounded height can't have all conjugates "close together",
        so by finite evil,
        any echoing point has some conjugate bounded away from 1 in modulus.

    "Tropical geometry" to study behaviour "near 0 and \infty" :
    Claim:
        Exist A,B and finitely many dim 1 cosets H_1+b_1,...,H_s+b_s such that
        for all x \in C(\C), if 
            L := || \log|x| || > A
        then for some i
             log|| (x / H_i) - (b_i / H_i) || <= -BL
    Proof:
        Else, by compactness, in an elementary extension R' of the real
        field \R there exists a point x\in X(C') (where C' is the complex
        field of R', C' = R'+iR' \supset \R+i\R = \C) such that

        (i) for some i, x_i > \R or x_i^{-1} > \R
        (ii) for all dim 1 subtori H, and all b \in \C,
            || x/H - b/H || > |x_i|^q for all i and all rationals q\neq 0.

        We work with the standard part map st : C' \maps \C and the
        corresponding valuation v.

        Recall the transcendental valuation inequality (TVI):
        for all y \in G_m^N(C')
            trd(y) >= trd(st(y)) + ld_\Q(v(y))

        By (i) and TVI, \ld_\Q(v(x))=1. So say v(x/H)=0, H a dim 1 torus, and
        let \alpha := st(x/H); by TVI again, \alpha \in \Qalg^{N-1}.

        Let \beta := x/H - \alpha.
        By (ii), v(\beta_i) < qv(x_j) for all i,j and q != 0.  (*)

        But \trd(\beta, x) = \trd(x) = 1,
        so by TVI, \ld_\Q(v(\beta), v(x)) = \ld_\Q(v(x)) = 1,
        so v(\beta) \in \spanof{v(x)}_\Q.

        So by (*), v(\beta_i) <= 0 for all i.

        But this contradicts st(\beta)=0.

    Now let x be n-echoing.
    Baker-Wüstholz effective "linear forms in logarithms" =>
        log||([n]x / H_i - b_i / H_i)|| is (if defined) bounded below
    proportionally to -deg(x)^11 log(n) and h(x), h(b_i).

    [also have to deal with case [n]x / H_i = b_i / H_i... but let's ignore
    that]

    Latter are bounded, so by claim, || log|[n]x| || is bounded below
    proportionally to -deg(x)^11 log(n).

    So by lower bound (after replacing x with a conjugate) on || log|x| ||,
        deg(x) >= c n^{1/11}