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1. Intersective sets for sparse sets of integers

   https://doi.org/10.1017/etds.2024.73  

   BIENVENU, PIERRE-YVES; GRIESMER, JOHN T; LE, ANH N; LÊ, THÁI HOÀNG (May 2025, Ergodic Theory and Dynamical Systems)

   Abstract For$$E \subset \mathbb {N}$$, a subset$$R \subset \mathbb {N}$$isE-intersectiveif for every$$A \subset E$$having positive relative density,$$R \cap (A - A) \neq \varnothing $$. We say thatRischromatically E-intersectiveif for every finite partition$$E=\bigcup _{i=1}^k E_i$$, there existsisuch that$$R\cap (E_i-E_i)\neq \varnothing $$. When$$E=\mathbb {N}$$, we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when$$E = \mathbb {P}$$, the set of primes, or other sparse subsets of$$\mathbb {N}$$. Among other things, we prove the following: (1) the set of shifted Chen primes$$\mathbb {P}_{\mathrm {Chen}} + 1$$is both intersective and$$\mathbb {P}$$-intersective; (2) there exists an intersective set that is not$$\mathbb {P}$$-intersective; (3) every$$\mathbb {P}$$-intersective set is intersective; (4) there exists a chromatically$$\mathbb {P}$$-intersective set which is not intersective (and therefore not$$\mathbb {P}$$-intersective). 

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2. Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

   https://doi.org/10.1007/s10957-023-02269-2  

   Luan, Nguyen Ngoc; Nam, Nguyen Mau; Thieu, Nguyen Nang; Yen, Nguyen Dong (November 2023, Journal of Optimization Theory and Applications)
3. Private Prediction Sets

   https://doi.org/10.1162/99608f92.16c71dad  

   Angelopoulos, Anastasios Nikolas; Bates, Stephen; Zrnic, Tijana; Jordan, Michael I. (April 2022, Harvard Data Science Review)
4. Sequences of Sets

   Benson, Austin R; Kumar, Ravi; Tomkins, Andrew. (April 2018, The 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD))

   Sequential behavior such as sending emails, gathering in groups, tagging posts, or authoring academic papers may be characterized by a set of recipients, attendees, tags, or coauthors respectively. Such łsequences of sets" show complex repetition behavior, sometimes repeating prior sets wholesale, and sometimes creating new sets from partial copies or partial merges of earlier sets. In this paper, we provide a stochastic model to capture these pat- terns. The model has two classes of parameters. First, a correlation parameter determines how much of an earlier set will contribute to a future set. Second, a vector of recency parameters captures the fact that a set in a sequence is more similar to recent sets than more distant ones. Comparing against a strong baseline, we ind that modeling both correlation and recency structures are required for high accuracy. We also ind that both parameter classes vary widely across domains, so must be optimized on a per-dataset basis. We present the model in detail, provide a theoretical examination of its asymptotic behavior, and perform a set of detailed experiments on its predictive performance. 

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5. Complementary Numerical Sets

   Guhl, Matthew; Juarez, Jazmine; Ponomarenko, Vadim; Rechkin, Rebecca; Singhal, Deepesh (January 2022, integers)