Here is the example of sums exhibiting a large amount of combinatorial cancellation that I mentioned in a comment on my previous post. First, some notation: p is a prime number, k is a positive integer, and G is the set of hyperplanes in the vector space Fpk of dimension k over the finite field with p elements. Then consider
$latex \sum_{\emptyset\not=I\subset \mathbf{G}}{\frac{(-1)^{|I|+1} }{1-p^{j(I)-k}}}$
where, for any subset I of G, we let
$latex j(I)=\mathrm{dim} \bigcap_{H\in I}{H}$
(the dimension of the intersection of the hyperplanes in I).
The terms of this sum are of absolute value of size roughly 1, if p is large (since I is non-empty, the intersection is contained in any of the hyperplanes in I, so j(I)<k), but there are many of them, namely
$latex 2^{|\mathbf{G}|}-1=2^{p^k+\cdots+p+1}-1,$
so one may suspect that the overall sum is quite large.
And yet, it turns out that, for fixed k and p going to infinity, the sum converges to k. My explanation for this is rather indirect, and I will explain it in another post soon, but maybe some readers can see a direct reason?
