arXiv subjects

Michael D. Boshernitzan

Publications and source records attributed to Michael D. Boshernitzan.

2 recordsLinked to original sources

arXiv2025

A cop-robber game on metric graphs

We study a variant of the classical cop-robber game played on compact metric graphs, where each edge is assigned a positive length and identified with a real interval of corresponding length. In this setting, both the cop and the robber move continuously along the edges, subject to upper bounds on their speeds. The cop has no knowledge of the robber's location and must choose a continuous path through the graph that is guaranteed to intersect the robber's trajectory at some point in time. We show that for every compact metric graph, there exists a constant s > 0 such that if the cop's speed exceeds s times the robber's speed, then the cop can guarantee capture.

math.CO↗

arXiv2024

Exceptional Points for Density Modulo 1

It is well known that almost every dilation of a sequence of real numbers, that diverges to $\infty$, is dense modulo~1. This paper studies the exceptional set of points -- those for which the dilation is not dense. Specifically, we consider the Hausdorff and modified box dimensions of the set of exceptional points. In particular, we show that the dimension of this set may be any number between 0 and 1. Similar results are obtained for two ``natural'' subsets of the set of exceptional points. Furthermore, the paper calculates the dimension of several sets of points, defined by certain constraints on their binary expansion.

math.NT↗