Francesco Cellarosi (Queen's University)

Department Colloquium

Speaker: Francesco Cellarosi (Queen's University)

Title: r-free integers and lattice points: dynamical, number theoretical, and probabilistic perspectives

Abstract: An integer is called r-free if it is not divisible by the r-th power of any prime. When r is at least 2, these integers have positive density but exhibit arbitrarily large gaps. The study of their distribution provides a rich playground for number theorists, dynamicists, probabilists, and AI enthusiasts. I will survey classical and recent results on squarefree and r-free integers and discuss their generalizations to higher-dimensional lattices and number fields.

From the dynamical perspective, I will describe my recent work on intrinsic ergodicity for the associated systems. I will then turn to percolation among squarefree Gaussian integers, focusing on the existence of paths to infinity with bounded step lengths and connections with OpenAI’s recent result on the Gaussian moat problem. Finally, I will discuss the frequencies of large gaps and holes and their probabilistic implications. For instance, we will discuss a conjecture on Poisson statistics for the occurrence of large gaps between squarefree integers, and (time permitting) the structure of the Voronoi tessellation generated by visible lattice points in the plane.