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Research

Publications and preprints

-[16] Density of shapes of periodic tori in the cubic case πŸ”— [arXiv] , with Thi Dang, Nihar Gargava, arXiv:2502.12754

-[15] Selberg, Ihara and Berkovich πŸ”— [arXiv], with Carlos Matheus, Wenyu Pan, Zhongkai Tao, arXiv:2412.20754

-[14] On the Dimension of Limit Sets on RP^2 via Stationary Measures II: variational principles and applications πŸ”— [arXiv] , with Yuxiang Jiao, Wenyu Pan, Disheng Xu, accepted by IMRN

-[13] On the dimension of limit sets on RP^2 via stationary measures I: the theory and Applications πŸ”— [arXiv], with Wenyu Pan, Disheng Xu, arXiv:2311.10265

-[12] Exponential mixing of frame flows for geometrically finite hyperbolic manifolds Β πŸ”— [arXiv], with Pratyush Sarkar, Wenyu Pan, arXiv:2302.03798, accepted by J. Eur. Math. Soc.

-[11] Stationary measures for SL2(ℝ)-actions on homogeneous bundles over flag varieties πŸ”— [arXiv], with Alex Gorodnik, Cagri Sert, accepted by Crelle’s Journal

-[10] Equidistribution and counting of periodic tori in the space of Weyl chambers, with Nguyen-Thi Dang πŸ”— [arXiv], accepted by Commentarii Mathematici Helvetici,

See arXiv:2202.08323 for an older version containing the case of SL(n,Z).

-[9] Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps πŸ”— [arXiv], with Wenyu Pan, Invent. math. 231, 931–1021 (2023).

-[8] Appendix of: The space of homogeneous probability measures on Ξ“βˆ–X is compact πŸ”— [arXiv], by Christopher Daw, Alexander Gorodnik, Emmanuel Ullmo, Math. Ann. 386, 987–1016 (2023).

-[7] Fourier transform of self-affine measures, with Tuomas Sahlsten πŸ”— [arXiv], Advances in Mathematics 374 (2020).

-[6] Trigonometric Series and Self-similar Sets, with Tuomas Sahlsten πŸ”— [arXiv], J. Eur. Math. Soc. 24 (2022), no. 1, pp. 341–368

[1], [6] and [7] use a similar idea, renewal theorem implies decay of Fourier transform of measures on fractal sets. [6] is the simplest non trivial case and is the easiest to read.

-[5] Kleinian Schottky groups, Patterson-Sullivan measures, and Fourier decay, with an appendix on stationarity of Patterson-Sullivan measures πŸ”— [arXiv], with Frédéric Naud and Wenyu Pan, Duke Math. J. 170, issue 4, (2021) pp. 775 - 825.

-[4] Fourier decay, Renewal theorem and Spectral gaps for random walks on split semisimple Lie groups πŸ”— [arXiv], Annales Scientifiques de l’ÉNS, Tome 55, Fasc.6, pp 1613-1686, 2022

[4] and [5] generalize the idea of Bourgain-Dyatlov, non-concentration and discretized sum-product estimates imply decay of Fourier transform of Furstenberg measures. A draft on SL_2(R) maybe helpful to understand the method. See also a post of Carlos Matheus for my talk on this topic.

-[3] Discretized Sum-product and Fourier decay in Rn πŸ”— [arXiv], Journal d’Analyse MathΓ©matique, 143, (2021) pp. 763–800.

-[2] Finiteness of Small Eigenvalues of Geometrically Finite Rank one Locally Symmetric Manifolds πŸ”— [arXiv], Mathematical Research Letters, 27, number 2 (2020) pp. 465 – 500.

-[1] Decrease of Fourier Coefficients of Stationary Measures πŸ”— [arXiv], Mathematische Annalen, 372, (2018) pp. 1189–1238.