Effros, Baire, Steinhaus and Non-Separability

Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus-Weil Theorem

The theme here is category-measure duality, in the context of a topological group. One can often handle the (Baire) category case and the (Lebesgue, or Haar) measure cases together, by working bi-topologically: switching between the original topology and a suitable refinement (a density topology). This prompts a systematic study of such density topologies, and the corresponding $σ$-ideals of negligibles. Such ideas go back to Weil's classic book, and to Hashimoto's ideal topologies. We make use of group norms, which cast light on the interplay between the group and measure structures. The Stei…

Generalization of two theorems of Steinhaus in category bases

Here we unify two results of Steinhaus and their corresponding category analogues by extending them in the settings of category bases. We further show that in any perfect translation base, every abundant Baire set contains a full subset for which our second theorem fails.

具有充分拓扑结构的集合中的点构型与一个拓扑Erdős相似性猜想

Point configurations in sets of sufficient topological structure and a topological {E}rdős similarity conjecture

该研究将Piccard拓扑结论推广至复杂点构型与抽象空间，证明非贫Baire集包含有界可数序列的缩放平移副本，验证了拓扑Erdős相似性猜想下有界可数集的通用性。