A structure sheaf for Kirch topology on N

Abstract: Kirch topology on $\mathbb N$ goes back to a 1969 paper of Kirch. It can be defined by a basis of open sets that consists of all infinite arithmetic progressions $a+d\mathbb N_0$, such that $\gcd(a,d)=1$ and $d$ is square-free. It is Hausdorff, connected, and locally connected. One can hope that in the classical imperfect analogy between arithmetic and geometry this can serve as an arithmetic analog of the usual topology on $\mathbb C$. However, the usual topology on $\mathbb C$ comes with a structure sheaf of complex-analytic functions. As far as I know, no analog for Kirch topology has been proposed before me. I believe that I have stumbled upon just such a thing, more by accident than by a conscious effort: locally LIP functions. These are functions from Kirch-open sets to $\mathbb Z$ such that for every point in the domain there is a Kirch-open neighborhood on which the function is "locally integer polynomial" (LIP): its interpolation polynomial on every finite set has integer coefficients. I will explain why this seems to be a natural object, what I know about it, and what I hope to achieve.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)