The separating Noether number βsep(G) of a finite group G is the minimal positive integer d such that for every finite G-module V, there is a separating set consisting of invariant polynomials of degree at most d. In this paper we obtain the exact value of the separating Noether number for all finite abelian groups via methods from additive combinatorics.
A subset Sof a poset Pis called a k-family if S is the union of k or fewer antichains of P. Greene and Kleitman showed that the set of k-families of a poset forms a locally distributive lattice. Kung, Rota, and Yan asked whether every finite locally distributive lattice can be represented in this way. It is shown here that the answer is no. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
A set B of positive integers is defined as a Sidon set if all differences b-b ' are distinct, where b, b ' E Band b =/ b '. By confirming a problem posed by Lev at CANT 2004, Cilleruelo and Nathanson (2008) [2] obtained a nice result as follows: for every Sidon set Band every function w(x)-* oo, there exists a set A of positive integers such that every positive integer can be uniquely expressed as a-a ', where a, a ' E A and A(x) >= B(x/3)-w(x) for all sufficiently large x. In this paper, we improve the coefficient from 1/3 to 1/2, that is: for every Sidon set Band every positive function w(x)-* oo, there exists a set A of positive integers such that every positive integer can be uniquely expressed as a-a ', where a, a ' E A and B(x/2)-w(x) <= A(x) <= B(x/2) + w(x) for all x >= 1. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Generalized Cayley maps are embeddings of Cayley graphs into closed surfaces, either orientable or non-orientable, admitting a group of automorphisms that acts regularly on the vertices. When the surface is orientable and the automorphism group preserves orientation, these embeddings, known as Cayley maps, have been extensively studied. A seminal result by Conder and Tucker classifies orientablyregular Cayley maps on cyclic groups. This paper completes the classification of regular generalized Cayley maps on cyclic groups by addressing the remaining cases: non-orientable surfaces or orientable surfaces with the regular cyclic group acting in an orientation-reversing manner. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
