The variant of remote set problem on lattices

Abstract

In 2015, Haviv proposed the Remote Set Problem (RSP) on lattices and gave a deterministic algorithm to find a set containing a point which is O(√k/n) far from the lattice in ℓp norm for 2 ≤ p≤∞, where n is the lattice rank and k divides n. Inspired by it, we propose the variant of Remote Set Problem on Lattices (denoted by V-RSP) that only depends on parameter γ ≤ 1. We obtain that the complexity classes that V-RSP belong to with the change of parameter γ. Using some elementary tools, we can solve V-RSP that can find a set containing a point which is O(k/n) far from the lattice in any ℓp norm for 1 ≤ p≤∞. Furthermore, we also study relationships between ℓ2 distance from a point to a lattice L and covering radius (ρ(p)(L)), where ρ(p)(L) is defined with respect to the ℓp norm for 1 ≤ p ≤ ∞, here, for p = ∞, our proof does not rely on Komlòs Conjecture.