Uri Shapira's homepage

Publications and Preprints:

(34) Escape of mass of the Thue-Morse sequence. Joint with Erez Nesharim and Noy Soffer Aranov. Submitted.

(33) On the rate of convergence of continued fraction statistics of random rationals. Joint with Ofir David, Ron Mor and Taehyeong Kim. To appear in Selecta Mathematica.

(32) Badly approximable grids and k-divergent lattices. Joint with Nikolay Moshchevitin and Anurag Rao. Mathematika, volume 70, issue 3, 2024.

(31) k-divergent lattices. Joint with Guy Lachman, Anurag Rau and Yuval Yifrach. Combinatorics and Number Theory (formerly Moscow Journal of Number Theory) Vol. 13 (2024), No. 3, 207–224 DOI: 10.2140/cnt.2024.13.207.

(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 Volume 201, pages 575–657, (2023).

(28) Translates of S-arithmetic orbits and applications. Joint with Cheng Zheng. Journal d'Analyse Mathématique. Volume 152, pages 137–180, (2024).

(27) Translates of rational points along expanding closed horocycles on the modular surface. Joint with Claire Burrin and Shucheng Yu. Mathematische Annalen. Volume 382, pages 655–717, (2022).

(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 (2020), 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 , Volume 40 , Issue 5 , May 2020 , pp. 1217 - 1237

(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, 2018, pages 149-176.

(21) Dimension bound for badly approximable grids. Joint with Seonhee Lim and Nicolas de Saxcé. International Mathematics Research Notices, Volume 2019, Issue 20, October 2019, Pages 6317–6346, https://doi.org/10.1093/imrn/rnx330.

(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. International Mathematics Research Notices. 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. International Mathematics Research Notices, Volume 2017, Issue 15, August 2017, Pages 4704–4731, https://doi.org/10.1093/imrn/rnw144

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

(14) Integer points on spheres and their orthogonal grids. Joint with Menny Aka and Manfred Einsiedler. Journal of the London Mathematical Society. (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. Compositio. Volume 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 (GGD). Volume 10, Issue 4, 2016, pp. 1211–1225.

(11) A volume estimate for the set of stable lattices. Joint with Barak Weiss. Comptes Rendus Mathématique (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. International Mathematics Research Notices, Volume 2015, Issue 14, 2015, Pages 5518–5559, https://doi.org/10.1093/imrn/rnu099.

(8) On the evolution of continued fraction expansions in a fixed quadratic field. Joint work with Menny Aka. Journal d'Analyse Mathématique. Volume 134, pages 335–397, (2018).

(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 of paper (6). There is a discussion about this mistake in the paper (32) Badly approximable grids and k-divergent lattices appearing above.

(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, pages 14–35, (2011).

(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.

(1) Measure Theoretical Entropy of Covers.  Israel Journal of math. Volume 158, pages 225–247, (2007).