Uri Shapira

Uri Shapira

Department of Mathematics
Technion – Israel Institute of Technology
Haifa, 32000

Phone: +972 (0)4 8293595
Room: Amado 705
Email: ushapira@gmail.com

I am a 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).
For more info about me see my CV.

I am currently on sabbatical at CU Boulder.

Publications and Preprints:

(31) k-divergent lattices. Joint with Guy Lachman, Anurag Rau and Yuval Yifrach. Submitted.

(30) Geometric and arithmetic aspects of approximation vectors. Joint with Barak Weiss. Submitted.

(29) Linnik’s problem in fiber bundles over quadratic homogeneous varieties. Joint with Michael Bersudsky. Monatshefte für Mathematik (2023) https://doi.org/10.1007/s00605-023-01836-z.

(28) Translates of S-arithmetic orbits and applications. Joint with Cheng Zheng. To appear in Journal d’Analyse Mathematique.

(27) Translates of rational points along expanding closed horocycles on the modular surface. Joint with Claire Burrin and Shucheng Yu. Mathematische Annalen. DOI:10.1007/s00208-021-02267-7.

(26) Limiting distributions of translates of divergent diagonal orbits. Joint with Cheng Zheng. Compositio Volume 155, Issue 9 2019 , pp. 1747-1793.

(25) On continued fraction expansions of quadratic irrationals in positive characteristic. Joint with Frédéric Paulin. Groups Geometry and Dynamics (GGD) vol 14, no. 1 pp. 81–105.

(24) Equidistribution of divergent orbits of the diagonal group in the space of lattices. Joint with Ofir David. Ergodic Theory and Dynamical Systems (ETDS). 18 pages. DOI: https://doi.org/10.1017/etds.2018.80

(23) Dynamics on the space of 2-lattices in 3-space. Joint with Oliver Sargent. GAFA (Geometric and Functional Analysis) volume 29, pages 890–948 (2019).

(22) Equidistribution of divergent orbits and continued fraction expansion of rationals. Joint with Ofir David. Journal of the London Mathematical Society (JLMS), volume 98, issue 1, pages 149-176.

(21) Dimension bound for badly approximable grids. Joint with Seonhee Lim and Nicolas de Saxcé. Int Math Res Notices, rnx330, https://doi.org/10.1093/imrn/rnx330. 25 pages.

(20) Asymptotic metric behaviour of random Cayley graphs. Joint with Reut Zuck. Combinatorica volume 39, pages 1133–1148 (2019).

(19) Dirichlet’s Shapes of unit lattices and escape of mass. Joint with Ofir David. Int Math Res Notices (IMRN) Vol. 2018, No. 9, pp. 2810–2843. doi: 10.1093/imrn/rnw324.

(18) Counterexamples to a conjecture of Woods. Joint with Oded Regev and Barak Weiss. Duke Math. J. 166 (2017), no. 13, 2443–2446. 4 pages.

(17) Escape of mass in homogeneous dynamics in positive characteristics. Joint with Alex Kemarsky and Frédéric 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.