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A local to global principle for expected values


Micheli, Giacomo; Schraven, Severin; Weger, Violetta (2022). A local to global principle for expected values. Journal of Number Theory, 238:1-16.

Abstract

This paper constructs a new local to global principle for expected values over free -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.

Abstract

This paper constructs a new local to global principle for expected values over free -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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Algebra and Number Theory
Uncontrolled Keywords:Algebra and Number Theory
Language:English
Date:1 September 2022
Deposited On:04 Oct 2022 13:56
Last Modified:27 Apr 2024 01:39
Publisher:Elsevier
ISSN:0022-314X
OA Status:Hybrid
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1016/j.jnt.2021.05.018
Project Information:
  • : FunderSNSF
  • : Grant ID200020_172623
  • : Project TitleDynamical and energetic properties of Bose-Einstein condensates
  • : FunderSNSF
  • : Grant ID200021_188430
  • : Project TitleNew Constructions of Convolutional Codes
  • : FunderSNSF
  • : Grant IDP2ZHP2_195290
  • : Project TitleTowards a Secure Post-Quantum World: New Metrics for Code-Based Cryptography
  • Content: Published Version
  • Licence: Creative Commons: Attribution 4.0 International (CC BY 4.0)