Dynamical Systems Seminar

In joint work with Kim and Marklof, we establish an estimate on the escape of mass in an average over a long unipotent segment in \(SL(2, \mathbb{Z})^3 \backslash SL(2, \mathbb{R})^3\) and apply this to the quantitative Oppenheim problem in shrinking intervals and the Berry-Tabor conjecture for 3-tori. The estimate is obtained by modifying Margulis function techniques developed by Eskin, Margulis, and Mozes in their resolution of the quantitative Oppenheim conjecture in fixed intervals. Our modification compensates for an exceptional set where certain locally defined averages do not decay, the contribution of which is controlled using techniques from analytic number theory combined with effective equidistribution results due to Lindenstrauss, Mohammadi, and Wang.