2025/01/08 by Yifrach, Yuval
#11J04 #37A05 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2501.04430
For every vector α∈ \RRn and for every rational approximation ( p,q)∈ \RRn×\RR we can associate the displacement vector qα- p. We focus on algebraic vectors, namely α=(α1,…,αn) such that 1, α1, …, αn span a rank n number field. For these vectors, we investigate the size of their displacements as well as the distribution of their directions. We give a new proof to the result of Bugeaud in \citeYannPAdic saying that algebraic vectors α satisfy the p-adic Littlewood Conjecture. Namely, we prove that \liminfk → ∞ ( k \abskp )1/n ‖ k (α1, …, αn) ‖_∞ = 0. Our new proof lets us classify all limiting distributions, with a special weighting, of the sequence of directions of the defects in the ε-approximations of (α1, …, αn). Each such limiting measure is expressed as the pushforward of an algebraic measure on Xn to the sphere.