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A note on the Möbius uncertainty principle for posets

Anurag Sahay

Abstract

We consider two generalizations of Pollack's uncertainty principle for Möbius inversion to locally finite posets. The first generalization was previously studied by Goh. Here, we provide a simplified sufficient criterion for the uncertainty principle to hold. We also provide a necessary criterion for the same which, in particular, disproves Goh's conjecture on the characterization of posets for which an uncertainty principle holds. The second generalization is new and applies to posets with reduced incidence algebras of a certain form. Here, we make some preliminary observations, including the fact that uncertainty principle holds for the poset of finite subsets of natural numbers and the poset of finite dimensional subspaces of $\mathbb{F}_q^\infty$. Our proofs in these settings are quite different from the proof for the poset of natural numbers under divisibility.

A note on the Möbius uncertainty principle for posets

Abstract

We consider two generalizations of Pollack's uncertainty principle for Möbius inversion to locally finite posets. The first generalization was previously studied by Goh. Here, we provide a simplified sufficient criterion for the uncertainty principle to hold. We also provide a necessary criterion for the same which, in particular, disproves Goh's conjecture on the characterization of posets for which an uncertainty principle holds. The second generalization is new and applies to posets with reduced incidence algebras of a certain form. Here, we make some preliminary observations, including the fact that uncertainty principle holds for the poset of finite subsets of natural numbers and the poset of finite dimensional subspaces of . Our proofs in these settings are quite different from the proof for the poset of natural numbers under divisibility.
Paper Structure (5 sections, 9 theorems, 52 equations, 2 figures)

This paper contains 5 sections, 9 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The sufficient condition
  3. The necessary condition
  4. Counterexamples
  5. The uncertainty principle for reduced incidence algebras

Key Result

Theorem 1.1

Let $f,g: \mathbb{N} \longrightarrow \mathbb{C}$ be arithmetic functions which are not identically zero such that Then, either $\mathop{\mathrm{supp}}\nolimits(f)$ or $\mathop{\mathrm{supp}}\nolimits(g)$ is infinite. Here, is the support of $f$.

Figures (2)

  • Figure 1: Schematic of the Hasse diagram for $P$.
  • Figure 2: Schematic of the Hasse diagram for $Q$.

Theorems & Definitions (19)

  • Theorem 1.1: Pollack
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Conjecture 1.6: Goh
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 2.1
  • ...and 9 more