Proves L is k-dense iff L_n,lim is G_delta and constructs k-dense lattices with sigma_n(L_n,lim) equal to 0 or 1 via Khintchine-Groshev theorem.
Smith Normal Form in Combinatorics
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abstract
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement.
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2026 1verdicts
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Topological and Diophantine properties of lattice subset projections
Proves L is k-dense iff L_n,lim is G_delta and constructs k-dense lattices with sigma_n(L_n,lim) equal to 0 or 1 via Khintchine-Groshev theorem.