2008/10/23 by Uri Shapira, Shapira, Uri · 4 citations
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics #Computer science #Conjecture #Diagonal #Diophantine equation #Discrete mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #Field (mathematics) #Geometry #Integer (computer science) #Mathematical Dynamics and Fractals #Mathematics #Number Theory (math.NT) #Pure mathematics #Span (engineering) #Type (biology) #math.DS #math.NT
paper · pdf · doi:10.48550/arxiv.0810.4289
published in arXiv (Cornell University) (Cornell University) · 12 pages
openalex publication_date 2008/10/23 · arxiv created 2009/05/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove that almost any pair of real numbers a,b, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (*) forall x,y in R, liminf|n|→∞ |n| = 0, where denotes the distance from the nearest integer. The existence of even a single pair that satisfies (*), solves a problem of Cassels from the 50's. We then prove that if 1,a,b span a totally real number field, then a,b, satisfy (*). It is further shown that if 1,a,b, are linearly dependent over Q, a,b cannot satisfy (*). The results are then applied to give examples of irregular orbit closures of the diagonal groups of a new type.