Simultaneous Approximations and Covering by Arithmetic Progressions in Fp

Given a set A={a1, …, an}⊆Fp of residues modulo prime p, we seek α, δ∈Fp (δ≠0) which simultaneously minimize all the distances ‖δai−α‖ from the zero residue and investigate the quantity mn=max|A|=n minα, δ max1⩽i⩽n ‖δai−α‖, the outer maximum being taken over all n-element subsets of Fp. It is shown that this extremal simultaneous approximation problem is equivalent to the combinatorial problem of finding minimal ln such that any set of n residues modulo p can be covered by an arithmetic progression of the length ln. For n⩾4, we determine the order of magnitude of mn and prove that 12p1−1/(n−1)(1+o(1))<mn<n−1/(n−1)p1−1/(n−1)(1+o(1)) (as p→∞ and n is small compared to p). For n=3, we find a sharp asymptotic and moreover, prove that −p/3<m3−p/3<1/2. These results answer a question of Straus about the maximum possible affine diameter of an n-element set of residues modulo a prime.