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1. Shrinking target problems for flows on homogeneous spaces

   https://doi.org/10.1090/tran/7783  

   Kelmer, Dubi; Yu, Shucheng (February 2019, Transactions of the American Mathematical Society)

   We study shrinking targets problems for discrete time flows on a homogeneous space Γ\G with G a semisimple group and Γ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows, settling a problem raised by Athreya and Margulis. 

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2. Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

   https://doi.org/10.3934/jmd.2021014  

   Kelmer, Dubi; Oh, Hee (January 2021, Journal of Modern Dynamics)

   Let \begin{document}$$ \mathscr{M} $$\end{document} be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets. 

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3. A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

   https://doi.org/10.1093/imrn/rnaa256  

   Kleinbock, Dmitry; Rao, Anurag (October 2020, International Mathematics Research Notices)

   Abstract We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $$\mathbb{R}^2$$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $$\psi $$, a zero-one law for the set of $$\alpha \in \mathbb{R}$$ such that for all large enough $$t$$ the inequality $$\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$$ has non-trivial integer solutions. 

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4. Zero–one laws for eventually always hitting points in rapidly mixing systems

   https://doi.org/10.4310/MRL.2023.v30.n3.a7  

   Kleinbock, Dmitry; Konstantoulas, Ioannis; Richter, Florian K (January 2023, Mathematical Research Letters)

   In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauß map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co-null set respectively). 

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5. An integer program for positive semidefinite zero forcing in graphs

   https://doi.org/10.1002/net.21947  

   Smith, Logan A.; Mikesell, Derek J.; Hicks, Illya V. (May 2020, Networks)

   Abstract Positive semidefinite (PSD) zero forcing is a dynamic graph process in which an initial subset of vertices are colored and may cause additional vertices to become colored through a set of color changing rules. Subsets which cause all other vertices to become colored are called PSD zero forcing sets; the PSD zero forcing number of a graph is the minimum cardinality attained by its PSD zero forcing sets. The PSD zero forcing number is of particular interest as it bounds solutions for the minimum rank and PSD min rank problems, both popular in linear algebra. This paper introduces blocking sets for PSD zero forcing sets which are used to formulate the first integer program (IP) for computing PSD zero forcing numbers of general graphs. It is shown that facets of the feasible region of this IP's linear relaxation correspond to zero forcing forts which induce connected subgraphs, but that identifying min cardinality connected forts is‐hard in general. Auxiliary IPs used to find these blocking sets are also given, enabling the master IP to be solved via constraint generation. Experiments comparing the proposed methods and existing algorithms are provided demonstrating improved runtime performance, particularly so in dense and sparse graphs. 

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