For t ∈[-1, 1), a set of points on the (n-1)-dimensional unit sphere is called t-almost equiangular if among any three distinct points there is a pair with inner product t. We propose a semidefinite programming upper bound for the maximum cardinality α(n, t) of such a set based on an extension of the Lovász theta number to hypergraphs. This bound is at least as good as previously known bounds and for many values of n and t it is better.
We also refine existing spectral methods to show that α(n, t) ≤2(n+1) for all n and t ≤0, with equality only at t = -1/n. This allows us to show the uniqueness of the optimal construction at t = -1/n for n ≤5 and to enumerate all possible constructions for n ≤3 and t ≤0.
@unpublished{bachoc2025obtusealmostequiangularsets,title={Obtuse almost-equiangular sets},year={2025},author={Bachoc, Christine and Bekker, Bram and Moustrou, Philippe and de Oliveira Filho, Fernando Mário},url={https://doi.org/10.48550/arXiv.2504.11086},doi={2504.11086},}
2023
Note
On the convergence of the k-point bound for topological packing graphs
We show that the k-point bound of de Laat, Machado, Oliveira, and Vallentin, a hierarchy of upper bounds for the independence number of a topological packing graph derived from the Lasserre hierarchy, converges to the independence number.
@unpublished{bekker2023convergence,title={On the convergence of the $k$-point bound for topological packing graphs},year={2023},author={Bekker, Bram and de Oliveira Filho, Fernando Mário},url={https://doi.org/10.48550/arXiv.2306.02725},doi={2306.02725},}
TAMS, to appear
Optimization hierarchies for distance-avoiding sets in compact spaces
Bram Bekker, Olga Kuryatnikova, Fernando Mário Olivera Filho, and Juan C. Vera
Transactions of the American Mathematical Society, 2023
Witsenhausen’s problem asks for the maximum fraction αn of the n-dimensional unit sphere that can be covered by a measurable set containing no pairs of orthogonal points. The best upper bounds for αn are given by extensions of the Lovász theta number. In this paper, optimization hierarchies based on the Lovász theta number, like the Lasserre hierarchy, are extended to Witsenhausen’s problem and similar problems. These hierarchies are shown to converge and are used to compute the best upper bounds for αn in low dimensions.
@article{Bekker2023OptimizationSpaces,title={{Optimization hierarchies for distance-avoiding sets in compact spaces}},year={2023},author={Bekker, Bram and Kuryatnikova, Olga and de Olivera Filho, Fernando Mário and Vera, Juan C.},journal={Transactions of the American Mathematical Society},url={https://doi.org/10.48550/arXiv.2304.05429},doi={2304.05429},}
Buildings are beautiful mathematical objects tying a variety of subjects in algebra and geometry together in a very direct sense. They form a natural bridge to visualizing more complex principles in group theory. As such, they provide an opportunity to talk about the inner workings of mathematics to a broader audience, but the visualizations could also serve as a didactic tool in teaching group and building theory, and we believe they can even inspire future research. We present an algorithmic method to visualize these geometric objects. The main accomplishment is the use of existing theory to produce three-dimensional, interactive models of buildings associated with groups with a BN-pair. The final product, an interactive web application called The Buildings Gallery, can be found at https://buildings.gallery/ [Bekker, B. (2021, June). The Buildings Gallery].
@article{Bekker2022TheBuildings,title={{The buildings gallery: visualizing buildings}},year={2022},journal={Journal of Mathematics and the Arts},author={Bekker, Bram and Solleveld, Maarten},month=apr,pages={1--18},url={https://www.tandfonline.com/doi/full/10.1080/17513472.2022.2063782},doi={10.1080/17513472.2022.2063782},issn={1751-3472},}