Singular matrices on fractals

Abstract: Singular vectors are those for which Dirichlet’s Theorem can be improved by arbitrarily small multiplicative constants. Recently, Kleinbock and Weiss showed that the set of singular vectors has measure zero with respect to any friendly measure. However, determining their Hausdorff dimension remains a subtle and challenging problem. Khalil addressed this by proving that the Hausdorff dimension of the set of singular vectors intersecting a self-similar fractal is strictly smaller than the fractal’s dimension. In this talk, I will extend Khalil’s result in four key directions. First, we generalize the study from vectors to matrices. Second, we analyze intersections with products of fractals, such as the Cartesian product of the middle-third and middle-fifth Cantor sets. Third, we establish upper bounds for singular vectors in a generalized weighted setting. Finally, we derive an upper bound on the Hausdorff dimension of $\omega$-very singular matrices in these broader settings, extending earlier work of Das, Fishman, Simmons, and Urbanski, who studied the real, unweighted case. Our approach is dynamical in nature, relying on the construction of a height function inspired by the work of Kadyrov, Kleinbock, Lindenstrauss, and Margulis. This is a joint work with Anish Ghosh.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar