Sorry Charles! It's an old chestnut though :-)

These are tempered distributions, and Andy's argument carries over verbatim to these.

There's less to the problem than this. Embed $L$ in $\mathbb{C}$ and consider the complex conjugation map $c$. Then $c$ restricts to an automorphism of $L$ (why?) and so corresponds to one of the elements of the group $Q$. Which elements of $Q$ are possible?

That's more-or-less the same problem as considering the first expected return of an asymmetric one-dimensional random walk to the origin.

I suspect Seb67 may assume that the "main diagonal" goes from $(0,0)$ to $(n,n)$. That's how I would interpret it. I'm not sure what "cross" means either: maybe three consecutive points $P_1$, $P_2$, $P_3$ with $P_2$ on the diagonal and $P_1$ and $P_3$ on opposite sides, or maybe just with $P_2$ on the diagonal.

Me too, Victor :-) I don't understand this anti-measure theory ideology either. I'm the first to want to avoid lengthy and technical proofs ('cos I can't understand them), but I prefer not to replace them with even more lengthy and even more technical proofs. :-)

So I should amend my original statement to "So, you are discarding the lengthy and technical proofs in measure theory in favour of Poincare duality and the de Rham theorem". (It is fortunate that these have short and non-technical proofs :-)). Alas, the proof of de Rham's theorem I have come closest to understanding is that in Warner's book. He relies on Poincare's Lemma (for the proof that the sheafified de Rham complex is exact) and for that he uses integration :-(

Neither does Hatcher's proof use de Rham cohomology.

So you are discarding the lengthy and technical proofs in measure theory in favour of Poincare duality. So, how does one construct the duality isomorphism $H^n_c(M,\mathbb{Or}(M))\to H_0(M)$ (presumably with real coefficients) without integration? I'd appreciate any hints/references (even if just for the case $M=\mathbb{R}^n$).

How about applying the Lefschetz fixed-point theorem?

Ah, QHLIU, you don't actually mean "proved", but have not removed the word from the original question.

QHLIU: have you a mathematical question here? Are you still asserting that the assertion you claimed was "proved" is proved? If so please can you give a reference?

QHLIU: so, when you said the statement was proved, that wasn't actually what you meant :-( Also now both $n$ and $N$ are "linearly indpendent" and $n/N$ is constant. :-(

QHLIU: in the original question, you said the limit was "proved". Now you say it's a "problem". What exacty do you mean?