The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-46954 (thanks to Qiaochu Yuan for pointing out the link).
*What I am really interested in is the description of the exceptional set. About how and where they are located ? Is there some kind of an estimate on how many of them are there ? conditions under which the exceptional set is empty / rather small / far from the origin etc. etc. in short any description about the geometry of the zero set. *
Here is an observation
Apparently all the known proofs use p-adic methods. Incidentally a lemma due to Turan comes very close to proving Skolem Mahler Lech theorem. It says if one knows the values of an exponential polynomial along an n length AP then one can find an estimate of it at the n+1 th point. Here is the exact statement
$\underline {Turan's Lemma}$ Let $ z_1,\dots,z_n$ be complex numbers, $|z_j|\geq 1, j=1,\dots,n.$ Let $ b_1,\dots, b_n \in \mathbb C $ and $$S_j:= \Sigma_{k=1}^n b_k z_k^j$$ Then $$|S_0| \leq \{\frac{4 e (m+n-1)}{n}\}^{n-1} \max_{j=m+1}^{m+n} |S_j|.$$ As a simple consequence of this result when the value of an exponential polynomial (with constant coefficient) is known for $n$ consecutive term of an arithmetic progression, then one can get an estimate of the value of the polynomial along that arithmetic progression. i.e., Let $p(t)=\Sigma_{k=1}^n c_k e^{i \lambda_k t}$ and assume that the value of the polynomial $p(t)$ is known for $t_j=t_0+j \delta$ for $ j= m+1,...,m+n$. Then substitute $b_k=c_k e^{i \lambda_j t_0}$ and $z_k= e^{i \lambda_k \delta}$ and apply Turan's lemma.
Applied to the the specific case under consideration ( i.e. an exponential polynomial of order n) it says if the exponential polynomial vanishes at an arithmetic progression of length n then it vanishes on the complete arithmetic progression.
So Szemeredi's Theorem now tells us that the structure of the zero set has to be union of complete AP plus a set of density zero set (actually more the exceptional set has no AP of length n).
Though it does not prove Skolem Mahler Lech theorem it's tantalizingly close and provides the extra information about the geometry of the zero sets .... that the exceptional set do not have an n length AP and using estimates from Ramsey theory one can find a bound on how many zeros can be there in an interval ?
Can one use all of these to give an analytic proof of Skolem Mahler Lech theorem maybe ? That would be real nice.
PS
Having said all that what is a good reference to what is known about the nature of the exceptional set. Can somebody suggest a good reference ? a good review article maybe.
My problem is though the statement about Skolem Mahler Lech Theorem seems like a problem in analysis/ complex analysis the relevant literature is in other fields about which I have very little familiarity. My present ambition is just to understand the results (not the proofs) that is what can one infer about the nature of the zero set (may be with some additional condition imposed on the set of exponents).
