A local to global principle for expected values

Giacomo Micheli,Severin Schraven,Violetta Weger

doi:10.1016/j.jnt.2021.05.018

Giacomo Micheli, Severin Schraven + Show 1 more

Open Access

https://doi.org/10.1016/j.jnt.2021.05.018

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Abstract

This paper constructs a new local to global principle for expected values over free Z-modules of finite rank. In our strategy we use the same philosophy as Ekedahl'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.

Highlights

Let Z be the set of integers and d be a positive integer
Since no uniform probability distribution exists over Z, one introduces the notion of density
Density results over Zd have received a great deal of interest recently [4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 22, 24]

Summary

Introduction

Let Z be the set of integers and d be a positive integer. The problem of computing the “probability”, that a randomly chosen element in Zd has a certain property, has a long history dating back to Cesaro [2, 3]. In [20, Lemma 20] Poonen and Stoll show that the computation of densities of many sets S ⊆ Zd defined by local conditions (in the p-adic sense) can be reduced to measuring the corresponding subsets of the p-adic integers. This technique is an extension of Ekedahl’s Sieve [23]. Stoll is not sufficient to guarantee a local to global principle for expected values (see Example 12) This led us to add two additional hypotheses on the system (Up)p∈P , that appears in the Poonen and Stoll principle, in order to prove Theorem 13, the main theorem of this article.

Preliminaries

The local to global principle for expected values

Applications