Equidistribution; Homogeneous spaces; Dynamical systems; Number theory; Effectiveness; Cartan action
Aka Menny, Einsiedler Manfred, Li Han, Mohammadi Amir (2020), On effective equidistribution for quotients of SL(d,ℝ), in
Israel Journal of Mathematics, 236(1), 365-391.
EINSIEDLER MANFRED, MAIER ALEX (2020), Simultaneous equidistributing and non-dense points for non-commuting toral automorphisms, in
Ergodic Theory and Dynamical Systems, 40(1), 175-193.
Einsiedler Manfred, Lindenstrauss Elon, Mohammadi Amir (2020), Diagonal actions in positive characteristic, in
Duke Mathematical Journal, 169(1), 117-175.
Einsiedler M., Margulis G., Mohammadi A., Venkatesh A. (2020), Effective equidistribution and property $(\tau )$, in
Journal of the American Mathematical Society, 33(1), 223-289.
Einsiedler Manfred, Lindenstrauss Elon (2019), Joinings of higher rank torus actions on homogeneous spaces, in
Publications mathématiques de l'IHÉS, 129(1), 83-127.
EINSIEDLER MANFRED, RÜHR RENÉ, WIRTH PHILIPP (2019), Distribution of shapes of orthogonal lattices, in
Ergodic Theory and Dynamical Systems, 39(06), 1531-1607.
Lytle Beverly, Maier Alex (2018), Simultaneous dense and nondense orbits for noncommuting toral endomorphisms, in
Monatsh. Math., 185(3), 473-488.
Einsiedler Manfred, Lindenstrauss Elon (2018), Symmetry of entropy in higher rank diagonalizable actions and measure classification, in
Journal of Modern Dynamics, 13(1), 163-185.
Einsiedler Manfred, Ghosh Anish, Lytle Beverly (2016), Badly approximable vectors, C^1 curves and number fields, in
Ergodic Theory Dynam. Systems, 36(6), 1851-1864.
Einsiedler Manfred, Mozes Shahar (2016), Divisibility properties of higher rank lattices, in
Transform. Groups, 21(4), 1039-1062.
Aka Menny, Einsiedler Manfred (2016), Duke's theorem for subcollections, in
Ergodic Theory Dynam. Systems, 36(2), 335-342.
Rühr Rene (2016), Effectivity of uniqueness of the maximal entropy measure on p-adic homogeneous spaces, in
Ergodic Theory Dynam. Systems, 36(6), 1972-1988.
Einsiedler Manfred, Mozes Shahar, Shah Nimish, Shapira Uri (2016), Equidistribution of primitive rational points on expanding horospheres, in
Compos. Math., 152(4), 667-692.
Aka Menny, Einsiedler Manfred, Shapira Uri (2016), Integer points on spheres and their orthogonal grids, in
J. Lond. Math. Soc. (2), 93(1), 143-158.
Aka Menny, Einsiedler Manfred, Shapira Uri (2016), Integer points on spheres and their orthogonal lattices, in
Invent. Math., 206(2), 379-396.
Einsiedler M., Kadyrov S., Pohl A. (2015), Escape of mass and entropy for diagonal flows in real rank one situations, in
Israel J. Math., 210(1), 245-295.
Einsiedler Manfred, Lindenstrauss Elon (2015), On measures invariant under tori on quotients of semisimple groups, in
Ann. of Math. (2), 181(3), 993-1031.
Badziahin Dmitry, Bugeaud Yann, Einsiedler Manfred, Kleinbock Dmitry (2015), On the complexity of a putative counterexample to the p-adic Littlewood conjecture, in
Compos. Math., 151(9), 1647-1662.
Bergelson Vitaly, Einsiedler Manfred, Tseng Jimmy (2015), Simultaneous dense and nondense orbits for commuting maps, in
Israel J. Math., 210(1), 23-45.
Equidistribution problems on homogeneous spaces are a central part of research in mathematics. One reason for the significance of these problems comes from the fact that they are intimately linked to different areas of mathematics. For instance the dynamical study of equidistribution problems on homogeneous spaces has become a powerful technique to address a variety of number theoretic problems; especially those involving Diophantine solutions to equations and inequalities - where the solution set and their equations both admit large symmetry groups.A very interesting equidistribution problem that has attracted a lot of attention since the work of Linnik from around 1960 is the distribution of integer points on large spheres with the origin at the center. Linnik established the equidistribution of the directions of these points unconditionally for d = 4 dimensions and under some mild congruence conditions for d = 3. Duke finally proved the case d = 3 unconditionally using subconvexity results on L-functions in 1988. A refinement of these results is to study the direction of the vector simultaneously with the shape of the lattice in the orthogonal complement. Once more the case of d = 3 is hardest, but as we explain in this proposal dynamics on homogeneous spaces can be used to analyze this problem for all dimensions d = 4 and also partially for d = 3.In the case of d = 4 the result by Mozes and Shah on the equidistribution of homogeneous measures is related to this problem. The Mozes-Shah result is a corollary of the theorems by Ratner on the dynamics of unipotent flows from around 1990. The theorems of Ratner, Mozes and Shah, Dani and Margulis in this area have found numerous applications to number theory and dynamics. Often this method leads to theorems that are unavailable by other means, but the disadvantage is that these results are non-effective. In a joint work the applicant, Margulis, and Venkatesh have obtained an effective equidistribution theorem by combining dynamical, number theoretical, and spectral methods. The theorem is currently restricted to the case of a real homogeneous space, and more importantly still has a technical (presumably avoidable) assumption that precludes some applications.More recently a second class of theorems have become useful for applications to number theory. These concern the dynamics of higher rank diagonalizable flows, and it is conjectured by Furstenberg, Margulis and others that these flows have similar rigidity properties as unipotent flows. The applicant has obtained in joint work with A. Katok and with E. Lindenstrauss a partial classification of positive entropy measures for higher rank diagonalizable flows. The assumption of positive entropy is crucial for the currently available techniques but even with this assumption there are still cases that have not been resolved and would have number theoretical applications.The main aim of this project is to extend the joint work of the applicant, Margulis, and Venkatesh as well as the work with E. Lindenstrauss and to apply these to number theoretic problems.