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On the expected value of energy in groups

Marco Barbieri, Marusa Lekše, Andoni Zozaya

Abstract

We obtain explicit upper and lower bounds for the expected action energy associated with a pair $({\sf A},{\sf Δ})$ of subsets sampled uniformly at random from a permutation group and its domain, respectively. We then specialize these bounds to multiplicative energy in several settings. In particular, we derive sharp asymptotic formulae for the expected energy of pairs of the form $({\sf A},{\sf A})$ and $({\sf A},{\sf A}^{-1})$. Finally, we apply these estimates to derive probabilistic results on the existence of subsets with large growth and to compare the typical behaviour of the cardinalities of the sets $|{\sf A}^{\ast 2}|$ and $|{\sf A}{\sf A}^{-1}|$.

On the expected value of energy in groups

Abstract

We obtain explicit upper and lower bounds for the expected action energy associated with a pair of subsets sampled uniformly at random from a permutation group and its domain, respectively. We then specialize these bounds to multiplicative energy in several settings. In particular, we derive sharp asymptotic formulae for the expected energy of pairs of the form and . Finally, we apply these estimates to derive probabilistic results on the existence of subsets with large growth and to compare the typical behaviour of the cardinalities of the sets and .
Paper Structure (26 sections, 29 theorems, 287 equations)

This paper contains 26 sections, 29 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. General bounds
  4. Finite groups
  5. Infinite groups
  6. Applications
  7. Acknowledgements
  8. Setup
  9. Notation
  10. Definition of energies
  11. Probability spaces and sampling
  12. Action energy
  13. Multiplicative energy of a set with itself
  14. Finite groups
  15. Infinite groups
  16. ...and 11 more sections

Key Result

Theorem 1

Let $G$ be a discrete group, let $\Omega$ be a discrete $G$-set, let $k,h: \mathbb{N} \to \mathbb{N}$ be two nondecreasing functions, let $\mathsf{A}_{k}$ be a $k$-subset of $F_n$ sampled uniformly at random, and let $\mathsf{\Delta}_{h}$ be a $h$-subset of $\Phi_n$ sampled uniformly at random. Then In particular, for every $\mathsf{A}_{k}$, $k$-subset of $G$ sampled uniformly at random, and for e

Theorems & Definitions (83)

  • Definition 1
  • Example 2
  • Example 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Theorem 1
  • proof
  • Lemma 7
  • proof
  • ...and 73 more