This paper provides a complete classification of GLn(R)-distinguished irreducible representations of GLn(& Copf;) when the representations are either generic or unitary. Additionally, for each such GLn(R)-distinguished representation, we explicitly construct the associated period and prove its non-vanishing on the distinguished minimal K-type excluding certain non-generic cases. Furthermore, we offer some applications to the branching problem using theta correspondence. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Given a prime p, the p-adic Littlewood Conjecture stands as a well-known arithmetic variant of the celebrated Littlewood Conjecture in Diophantine Approximation. In the same way as the latter, it admits a natural function field analogue depending on the choice of an irreducible polynomial P(t) with coefficients in a field K. This analogue is referred to as the P(t)-adic Littlewood Conjecture (P(t)-LC for short). P(t)-LC is proved to fail for any choice of irreducible polynomial P(t) over any ground field K with characteristic & ell; = 3 (mod 4). The counterexample refuting it is shown to present a local arithmetic obstruction emerging from the fact that-1 is not a quadratic residue modulo a prime & ell; = 3 (mod 4). The theory developed elucidates and generalises all previous approaches towards refuting the conjecture. They were all based on the computer-assisted method initiated by Adiceam, Nesharim and Lunnon (2021) which has been able to establish that P(t)-LC fails in some small characteristics (essentially up to 11). This computer-assisted method is, however, unable to provide a general statement as it relies on ad hoc computer verifications which, provided they terminate, refute P(t)-LC in a given characteristic. This limitation is overcome by exhibiting an arithmetic obstruction to the validity of P(t)-LC in infinitely many characteristics. The existence of arithmetic obstructions within the context of P(t)-LC leaves the remaining case of odd characteristics l 1 (mod4) dependent on their determination. This is shown to hold in an effective and explicit way. (c) 2026 The Author(s). Published by Elsevier Inc.
We give a categorical explanation for many properties of profinite coproducts of profinite groups, which were previously proven on a case-by-case basis. All of these properties take the form "certain functors preserve profinite coproducts". We give various examples to show how our framework can be applied. We also point out connections to internal category theory and profinite Bass-Serre theory. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
We introduce a topological notion of the size of subsets of the real line, called super-density. A set A C & Ropf; is superdense if for every non-empty open interval I C & Ropf; and every nowhere constant continuous function phi: I-* & Ropf;, we have phi(I fl A) fl A =/ 0. We first establish basic properties of superdense sets. Our main topological result characterizes them within the framework of Baire category: a set with the Baire property is super-dense if and only if it is co-meager. We then investigate the implications for the theory of normal numbers. Our topological considerations imply that no nowhere constant continuous function can map all non-normal numbers to normal numbers. Conversely, we explicitly construct a nowhere constant continuous function that maps all normal numbers to non-normal numbers. Finally, we provide a constructive procedure that, given any countable family of nowhere constant continuous functions, produces a real number x such that all its images under these functions are non-normal. As a corollary, we obtain the existence of a computable non-normal number x such that e alpha x is non-normal for every non-zero algebraic alpha. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
