Alladi Ramakrishnan Hall

Effective Quantitative Oppenheim for almost every quadratic form

Jayadev Athreya

University of Illinois at Urbana-Champaign

In 1986, Margulis, using methods from dynamics, proved A. Oppenheim's 1929 conjecture that for every indefinite irrational quadratic form in at least 3 variables, the values it takes at integer lattice points form a dense subset of the real line. Subsequently, Eskin-Margulis-Mozes proved an associated counting result, giving polynomial asymptotics for the number of lattice points of norm at most T which get mapped to a fixed interval. In joint work with Margulis, we give an effective version if this result for almost every quadratic form.