Rational Bass Note Spectra and Orbits in Homogeneous Spaces

In this talk, we will define the rational bass note spectrum of a homogeneous polynomial. We will explain how in certain special cases, its behavior is linked to heights of orbits inside homogeneous spaces, as well as many classical problems in simultaneous Diophantine approximation. We will describe the behavior of these orbits and as a consequence, we will prove the existence of infinite families of primitive integer binary cubic forms, so that the smallest natural number they represent non-trivially is surprisingly large.