Almost optimum ℓ-covering of ℤ_n
share
A subset B of ring ℤ_n is called a ℓ-covering set if { ab n | 0≤ a ≤ℓ, b∈ B} = ℤ_n. We show there exists a ℓ-covering set of ℤ_n of size O(n/ℓlog n) for all n and ℓ, and how to construct such set. We also show examples where any ℓ-covering set must have size Ω(n/ℓlog n/loglog n). The proof uses a refined bound for relative totient function obtained through sieve theory, and existence of a large divisor with linear divisor sum.
READ FULL TEXT