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1. Leibniz International Proceedings in Informatics (LIPIcs):18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

   https://doi.org/10.4230/LIPIcs.SWAT.2022.26  

   Gibson-Lopez, Matt; Zamarripa, Serge (January 2022, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Czumaj, Artur; Xin, Qin (Ed.)

   Representation of Euclidean objects in a digital space has been a focus of research for over 30 years. Digital line segments are particularly important as other digital objects depend on their definition (e.g., digital convex objects or digital star-shaped objects). It may be desirable for the digital line segment systems to satisfy some nice properties that their Euclidean counterparts also satisfy. The system is a consistent digital line segment system (CDS) if it satisfies five properties, most notably the subsegment property (the intersection of any two digital line segments should be connected) and the prolongation property (any digital line segment should be able to be extended into a digital line). It is known that any CDS must have Ω(log n) Hausdorff distance to their Euclidean counterparts, where n is the number of grid points on a segment. In fact this lower bound even applies to consistent digital rays (CDR) where for a fixed p ∈ ℤ², we consider the digital segments from p to q for each q ∈ ℤ². In this paper, we consider families of weak consistent digital rays (WCDR) where we maintain four of the CDR properties but exclude the prolongation property. In this paper, we give a WCDR construction that has optimal Hausdorff distance to the exact constant. That is, we give a construction whose Hausdorff distance is 1.5 under the L_∞ metric, and we show that for every ε > 0, it is not possible to have a WCDR with Hausdorff distance at most 1.5 - ε. 

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2. A dichotomy phenomenon for bad minus normed Dirichlet

   https://doi.org/10.1112/mtk.12221  

   Kleinbock, Dmitry; Rao, Anurag (August 2023, Mathematika)

   Abstract Given a norm ν on , the set of ν‐Dirichlet improvable numbers was defined and studied in the papers (Andersen and Duke,Acta Arith. 198 (2021) 37–75 and Kleinbock and Rao,Internat. Math. Res. Notices2022 (2022) 5617–5657). When ν is the supremum norm, , where is the set of badly approximable numbers. Each of the sets , like , is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either or else has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether thecritical locusof ν intersects a precompact ‐orbit, where is the one‐parameter diagonal subgroup of acting on the spaceXof unimodular lattices in . Thus, the aforementioned dichotomy follows from the following dynamical statement: for a lattice , either is unbounded (and then any precompact ‐orbit must eventually avoid a neighborhood of Λ), or not, in which case the set of lattices inXwhose ‐trajectories are precompact and contain Λ in their closure has full Hausdorff dimension. 

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3. Non-dense orbits on homogeneous spaces and applications to geometry and number theory

   https://doi.org/10.1017/etds.2021.4  

   AN, JINPENG; GUAN, LIFAN; KLEINBOCK, DMITRY (April 2022, Ergodic Theory and Dynamical Systems)

   Abstract Let G be a Lie group, let $$\Gamma \subset G$$ be a discrete subgroup, let $$X=G/\Gamma $$ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $$x\in X$$ with f -trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X . This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points. 

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4. Hausdorff Dimension of Caloric Measure

   https://doi.org/10.1353/ajm.2025.a954648  

   Badger, Matthew; Genschaw, Alyssa (April 2025, American Journal of Mathematics)

   abstract: We examine caloric measures $$\omega$$ on general domains in $$\RR^{n+1}=\RR^n\times\RR$$ (space $$\times$$ time) from the perspective of geometric measure theory. On one hand, we give a direct proof of a consequence of a theorem of Taylor and Watson (1985) that the lower parabolic Hausdorff dimension of $$\omega$$ is at least $$n$$ and $$\omega\ll\Haus^n$$. On the other hand, we prove that the upper parabolic Hausdorff dimension of $$\omega$$ is at most $$n+2-\beta_n$$, where $$\beta_n>0$$ depends only on $$n$$. Analogous bounds for harmonic measures were first shown by Nevanlinna (1934) and Bourgain (1987). Heuristically, we show that the \emph{density} of obstacles in a cube needed to make it unlikely that a Brownian motion started outside of the cube exits a domain near the center of the cube must be chosen according to the ambient dimension. In the course of the proof, we give a caloric measure analogue of Bourgain's alternative: for any constants $$0<\epsilon\ll_n \delta<1/2$ and closed set $$E\subset\RR^{n+1}$$, either (i) $$E\cap Q$$ has relatively large caloric measure in $$Q\setminus E$$ for every pole in $$F$$ or (ii) $$E\cap Q_*$$ has relatively small $$\rho$$-dimensional parabolic Hausdorff content for every $$n<\rho\leq n+2$$, where $$Q$$ is a cube, $$F$$ is a subcube of $$Q$$ aligned at the center of the top time-face, and $$Q_*$$ is a subcube of $$Q$$ that is close to, but separated backwards-in-time from $$F$$: \begin{gather*} Q=(-1/2,1/2)^n\times (-1,0),\quad F=[-1/2+\delta,1/2-\delta]^n\times[-\epsilon^2,0),\\[2pt] \text{and }Q_*=[-1/2+\delta,1/2-\delta]^n\times[-3\epsilon^2,-2\epsilon^2]. \end{gather*} Further, we supply a version of the strong Markov property for caloric measures. 

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5. Polyhedral approximation and uniformization for non-length surfaces

   https://doi.org/10.4171/JEMS/1538  

   Ntalampekos, Dimitrios; Romney, Matthew (September 2024, Journal of the European Mathematical Society)

   We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2-sphere with finite Hausdorff 2-measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala–Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk–Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain. 

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