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FirstView articles

Contents

Research Article

  • The Reidemeister Spectrum of Finite Abelian Groups

  • Part of:
  • Pieter Senden
  • Published online by Cambridge University Press: 06 September 2023, pp. 1-20

  • For a finite abelian group A, the Reidemeister number of an endomorphism φ is the same as the number of fixed points of φ, and the Reidemeister spectrum of A is completely determined by the Reidemeister spectra of its Sylow p-subgroups. To compute the Reidemeister spectrum of a finite abelian p-group P, we introduce a new number associated to an automorphism ψ of P that captures the number of fixed points of ψ and its (additive) multiples, we provide upper and lower bounds for that number, and we prove that every power of p between those bounds occurs as such a number.

  • Perron’s capacity of random sets

  • Part of:
  • Anthony Gauvan
  • Published online by Cambridge University Press: 01 September 2023, pp. 1-11

  • We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $\left\{X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions

    \begin{equation*}\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{equation*}
    and
    \begin{equation*}\Omega_{\text{rand},\text{lac}} := \left\{\frac{\pi X_k}{2^k} : k\geq 1 \right\}.\end{equation*}

    We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({\mathbb{R}}^2)$ for any $1 \lt p \lt \infty$.