Local to global principle for expected values

TL;DR

This paper develops a new local to global principle for expected values over free b-modules, extending density-based methods to expected value computations with sharp conditions and practical applications.

Contribution

It introduces a novel local to global principle for expected values, extending existing density methods and establishing sharp conditions for their validity.

Findings

Established a new local to global principle for expected values.

Identified sharp conditions necessary for the principle to hold.

Provided new applications and simplified proofs for known results.

Abstract

This paper constructs a new local to global principle for expected values over free $\mathbb{Z}$-modules of finite rank. In our strategy we use the same philosophy as Ekedhal's Sieve for densities, later extended and improved by Poonen and Stoll in their local to global principle for densities. We show that under some additional hypothesis on the system of $p$-adic subsets used in the principle, one can use $p$-adic measures also when one has to compute expected values (and not only densities). Moreover, we show that our additional hypotheses are sharp, in the sense that explicit counterexamples exist when any of them is missing. In particular, a system of $p$-adic subsets that works in the Poonen and Stoll principle is not guaranteed to work when one is interested in expected values instead of densities. Finally, we provide both new applications of the method, and immediate proofs for known results.