$$\frac{1}{\mu(B)}\int_B \vert x \vert d\mu(x) = \frac{1}{p+1}$$
This formula holds for the unit ball in $\mathbb{Q_p}$. This formula also holds for
$\mathbb{R}$ when $p=1$. Should one expect $$\mathrm{Frac}(W_{1^{\infty}} (\mathbb{F_1}))=\mathbb{R}?$$
What (mathematical) criteria do people use to rule-out field with one element phenomena? What makes point counting formulas better (or worse)?
Best Answer
It is true that if $f(T)\in \mathbb Z[T]$ then,
$$ \frac{1}{\mu_p(B)}\int_{\mathbb Z_p} f(\vert x \vert_p) d\mu_p(x) \to \frac{1}{\vert B \vert}\int_{B}f(\vert x \vert)dx \mbox{ as } p\to 1.$$ It is not in general true that $$ \frac{1}{\mu_p(B)}\int_{\mathbb Z_p} \vert f( x )\vert_p d\mu_p(x) \to \frac{1}{\vert B \vert}\int_{B}\vert f( x )\vert dx \mbox{ as } p\to 1.$$
When $f(x) = x^2-1$ we have $$\int_{\mathbb Z_p} \vert x^2-1 \vert_p d\mu(x) = \frac{1+p(p-2)}{p}+ \frac{1}{(p+1)p} \to 1/2 \mbox{ as } p \to 1 $$ and $$\frac{1}{2}\int_{-1}^1 \vert x^2-1\vert dx = 2/3.$$ http://imgur.com/a/m6KYA
A couple remarks:
You should have also written $\mathbb Q_1 = \mathbb{R}$ instead of that terrible notation $\mathrm{Frac}(W_{1^{\infty}}(\mathbb{F}_1))$.
The question as posed is a little stupid since if someone had a procedure for "ruling our phenomena" not only would they would probably have category in mind, but they would be able to compute with it. I think the correct answer is $\mathbb F_1$ numerology is justified when it can be categorified''. This is kind of a weak version of the statement "a conjecture is true when you can prove it". I guess a vague question deserves a vague answer.
The first part of the question seems to be about Arakelov geometry and replacing the place at infinity with the place 1. It seems that the answer is no. Another bad question would be to ask for more examples where it does make sense.
The last part of the original question can be made more precise:
-Are there examples of formulas for $|X(\mathbb F_q)|$ such that $X$ is definable in either monoidal algebraic geometries or Borger's $\Lambda$-ring categorifications of schemes over $\mathbb F_1$ such that the point counting formula's in the category do not agree with $|X(\mathbb F_q)|$ as $q\to 1$? (if you know of another categorification that justifies $GL_n(\mathbb{F}_1)=S_n$ this question applies there too (Maybe Durov's Category?).)