Jakub Byszewski

Simultaneous p-orderings and minimizing volumes in number fields

Abstract In [VP] , V.V. Volkov and F.V. Petrov consider the problem of existence of the so-called n-universal sets (related to simultaneous p -orderings of Bhargava) in the ring of Gaussian integers. A related problem concerning Newton sequences was considered by D. Adam and P.-J. Cahen in [AC] . We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov–Petrov on minimal cardinality of n-universal sets. Along the way, we discover a link with Euler–Kronecker constants and prove a lower bound on Euler–Kronecker constants which is of the same order of magnitude as the one obtained by Ihara.

Transitive dendrite map with zero entropy

Hoehn and Mouron [Ergod. Th. \& Dynam. Sys. (2014) \textbf{34}, 1897--1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn-Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Topology (2001) \textbf{40}, 551--569].