In this talk we present an extension of a probabilistic result of Marcus, Spielman, and Srivastava, which resolved the Kadison-Singer problem, for block diagonal positive semidefinite random matrices. We use this result to show several selector results, which generalize their partition counterparts. This includes a selector form of Weaver’s $\mathrm{KS}_r$ conjecture for block diagonal trace class operators, which extends a selector result for Bessel sequences, or equivalently rank one matrices, due to Londner and the author. We also show a selector variant of Feichtinger’s conjecture for a (possibly infinite) collection of Bessel sequences, extending earlier results for a single Bessel sequence. We prove a generalization of the $\mathrm{R}_\epsilon$ conjecture of Casazza, Tremain, and Vershynin for infinite collection of equal norm Bessel sequences. In particular, our selector result yields a conjectured asymptotically optimal bound for a single Bessel sequence in terms of Riesz sequence tightness parameter.

We consider vague limit of properly normalized zero counting measures of orthogonal polynomials on the real line corresponding to measures with unbounded support. One important feature is that we do not scale the argument of the polynomials and as a consequence we have to deal with lack of tightness of the corresponding sequence of measures. The limiting measure turns out to be an infinite Radon measure (for Hermite polynomials it is a constant multiple of the Lebesgue measure). This is a joint work with Bartosz Trojan (Polish Academy of Science).