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   https://doi.org/10.19086/aic.2025.1  

   Bukh, Boris; Zichao, Dong (January 2025, Advances in Combinatorics)

   For a finite point set P⊂R^d, denote by diam(P) the ratio of the largest to the smallest distances between pairs of points in P. Let c_{d,α}(n) be the largest integer c such that any n-point set P⊂R^d in general position, satisfying diam(P)<αn^{1/d}, contains an c-point convex independent subset. We determine the asymptotics of c_{d,α}(n) as n→∞ by showing the existence of positive constants β=β(d,α) and γ=γ(d) such that βn^{(d−1)/(d+1)}≤c_{d,α}(n)≤γn^{(d−1)/(d+1)} for α≥2. 

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2. Simplices and sets of positive upper density in $$\mathbb {R}^d$$

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   Huckaba, Lauren; Lyall, Neil; Magyar, Ákos (June 2017, Proceedings of the American Mathematical Society)
3. Falconer’s $(K, d)$ distance set conjecture can fail for strictly convex sets $$K$$ in $$\mathbb R^d$$

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   Bishop, Christopher; Drillick, Hindy; Ntalampekos, Dimitrios (January 2021, Revista Matemática Iberoamericana)
4. Product of Simplices and sets of positive upper density in ℝ ^d

   https://doi.org/10.1017/S0305004117000184  

   LYALL, NEIL; MAGYAR, ÁKOS (July 2018, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We establish that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δ k 1 and Δ k 2 are two fixed non-degenerate simplices of k 1 + 1 and k 2 + 1 points respectively, then any subset of ℝ d of positive upper Banach density with d ⩾ k 1 + k 2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δ k 1 × Δ k 2 . A new direct proof of the fact that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented. 

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5. Clustering characteristic diffraction vectors in 4-D STEM data sets from overlapping structures in nanocrystalline and amorphous materials

   https://doi.org/10.1016/j.ultramic.2024.114040  

   Francis, Carter; Voyles, Paul M (December 2024, Ultramicroscopy)

   We describe a method for identifying and clustering diffraction vectors in four-dimensional (4-D) scanning transmission electron microscopy data to determine characteristic diffraction patterns from overlapping structures in projection. First, the data is convolved with a 4-D kernel, then diffraction vectors are identified and clustered using both density-based clustering and a metric that emphasizes rotational symmetries. The method works well for both crystalline and amorphous samples and in high- and low-dose experiments. A simulated dataset of overlapping aluminum nanocrystals provides performance metrics as a function of Poisson noise and the number of overlapping structures. Experimental data from an aluminum nanocrystal sample shows similar performance. For an amorphous Pd77.5Cu6Si16.5 thin film, experiments measuring glassy structure show strong evidence of 4- and 6-fold symmetry structures. A significant background arises from the diffraction of overlapping structures. Quantifying this background helps to separate contributions from single, rotationally symmetric structures vs. apparent symmetries arising from overlapping structures in projection. 

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