Affine Diameter of a Set of Residues The affine diameter l(A) of a set A of residues modulo a prime number p was introduced by E.G.Straus in 1976; it is the length of the shortest arithmetic progression modulo p, containing A as its subset. For example, the affine diameter of A={1, 5, 6} (mod 7) is 3, the shortest progression being {6, 1, 3, 5}. (Actually, Straus used a slightly different definition and another notation. In our definition above, the length of a progression is the number, less by one than the number of its elements.) Straus raised the problem of estimating l(n), the maximum possible diameter of an n-element set of residues modulo given p. This problem is essentially solved in my recent paper "Simultaneous Approximations and Covering by Arithmetic Progressions modulo a prime", where I prove that As a next step, it is natural to ask for the sharp asymptotic for l(n) / p^{1-µ} or more precisely, for the liminf and limsup of this ratio as p grows to infinity and n is fixed. In the above mentioned paper, I was able to find the asymptotic for n=3 and moreover, to show that Conjecture. |