Approximately Jumping Towards the Origin

2024/12/05 by Alex Albors, François Clément, Albors, Alex +9 · 1 citation
Computer Science · #Cellular Automata and Applications

paper · pdf · doi:10.48550/arxiv.2412.04284

Abstract

Given an initial point x₀ ∈ ℝ^d and a sequence of vectors v₁, v₂, … in ℝ^d, we define a greedy sequence by setting x_n = x_n-1 ± v_n where the sign is chosen so as to minimize ‖x_n‖. We prove that if the vectors v_i are chosen uniformly at random from \mathbbS^d-1 then elements of the sequence are, on average, approximately at distance ‖x_n‖ ∼ √(πd/8) from the origin. We show that the sequence (‖x_n‖)_n=1^∞ has an invariant measure π_d depending only on d and we determine its mean and study its decay for all d. We also investigate a completely deterministic example in d=2 where the v_n are derived from the van der Corput sequence. Several additional examples are considered.

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