Department of Mathematics

Technion – Israel Institute of Technology

Haifa, 32000

Israel

Phone: +972 (0)4 8293595

Room: Amado 705

Email: ushapira@gmail.com

I am an associate professor at the mathematics department at the Technion.

My areas of interest are Homogeneous Dynamics, Ergodic Theory, and applications to Number Theory (especially to Diophantine approximation and the geometry of numbers).

**Seminars:**

Here is a link for the homepage of our weekly Groups, Dynamics, and Related Topics seminar (also known as GDRT seminar).

**Publications and Preprints:**

**(24)** Equidistribution of divergent orbits of the diagonal group in the space of lattices. Joint with Ofir David. Submitted. 18 pages.

**(23)** Dynamics on the space of 2-lattices in 3-space. Joint with Oliver Sargent. Submitted. 47 pages.

**(22)** Equidistribution of divergent orbits and continued fraction expansion of rationals. Joint with Ofir David. Submitted. 32 pages.

**(21)** Dimension bound for badly approximable grids. Joint with Seonhee Lim and Nicolas de Saxcé. Submitted. 25 pages.

**(20)** Asymptotic metric behaviour of random Cayley graphs. Joint with Reut Zuck. To appear in Combinatorica. 12 pages.

**(19)** Dirichlet’s Shapes of unit lattices and escape of mass. Joint with Ofir David. Int Math Res Notices 2017 rnw324. doi: 10.1093/imrn/rnw324. 30 pages.

**(18)** Counterexamples to a conjecture of Woods. Joint with Oded Regev and Barak Weiss. To appear in Duke. 3 pages.

**(17)** Escape of mass in homogeneous dynamics in positive characteristics. Joint with A. Kemarsky and F. Paulin. Journal of Modern Dynamics Volume 11, Issue 03, 2017 Pages 369-407.

**(16)** Full escape of mass for the diagonal group. Int Math Res Notices 2016 rnw144. doi: 10.1093/imrn/rnw144. 26 pages.

**(15)** Integer points on spheres and their orthogonal lattices. Joint with Menny Aka and Manfred Einsiedler. Invent. Math. 206 (2016), no. 2, 379–396.

**(14)** Integer points on spheres and their orthogonal grids. Joint with Menny Aka and Manfred Einsiedler. J. Lond. Math. Soc. (2) 93 (2016), no. 1, 143–158.

**(13)** Equidistribution of primitive rational points on expanding horospheres. Joint with Manfred Einsiedler, Shahar Mozes and Nimish Shah. Compos. Math. 152 (2016), no. 4, 667-692.

**(–)**. On stable lattices and the diagonal group. Joint with Barak Weiss.

**Remark:** Following the referees advise this article was split into three shorter articles appearing below.

**(12)**. Closed orbits for the diagonal group and Well-Rounded lattices. Joint with Barak Weiss and Michael Levin. Groups Geometry and Dynamics. Volume 10, Issue 4, 2016, pp. 1211–1225.

**(11)** A volume estimate for the set of stable lattices. Joint with Barak Weiss. R. Math. Acad. Sci. Paris 352 (2014), no. 11, 875-879.

**(10)** Stable lattices and the diagonal group. Joint with Barak Weiss. Journal of the European Mathematical Society. Volume 18, Issue 8, 2016, pp. 1753 – 1767.

**(9)** On the Mordell-Gruber spectrum. Joint work with Barak Weiss. Int. Math. Res. Not. IMRN 2015, no. 14, 5518-5559.

**(8)** On the evolution of continued fraction expansions in a fixed quadratic field. Joint work with Menny Aka. To appear in Journal d’analyse Mathematique. 53 pages.

**(7)** Remarks on Euclidean Minima. Joint work with Zhiren Wang. Journal of Number Theory 137 (2014) 93 – 121.

**(6)** Grids with dense values. Commentarii Mathematici Helvetici, Volume 88, Issue 2, 2013, 21 pages. Updated on 4/4/2011.

** Remark: ** David Simmons spotted a mistake in the published version. Hopefully, in the not-so-far away future, I shall have time to write a proper errata.

**(5)** Homogeneous orbit closures and applications. Joint work with E. Lindenstrauss. Ergodic Theory and Dynamical Systems, (a volume in memory of Dan Rudolph), Volume 32, Issue 02, April 2012, 22 pages.

**(4)** Diophantine approximations on fractals. Joint work with M. Einsiedler and L. Fishman. Geometric and Functional Analysis, Volume 21, Number 1, 21 pages.

**(3)** A solution to a problem of Cassels and Diophantine properties of cubic numbers. Annals of Mathematics Volume 173 (2011), no. 1, 14 pages.

**(2)** On a generalization of Littlewood’s conjecture. Journal of Modern Dynamics Volume 3, No. 3, 2009, 20 pages.

**(1)** Measure Theoretical Entropy of Covers. Israel Journal of math. 158(2007), (1). 22 pages.