A Note on a Problem of Hilliker and Straus

For a prime p and a vector ᾱ = (α1, . . . , αk) ∈ Zp let f (ᾱ, p) be the largest n such that in each set A ⊆ Zp of n elements one can find x which has a unique representation in the form x = α1a1 + · · · + αkak, ai ∈ A. Hilliker and Straus [2] bounded f (ᾱ, p) from below by an expression which contained the L1-norm of ᾱ and asked if there exists a positive constant c (k) so that f (ᾱ, p) > c (k) log p. In this note we answer their question in the affirmative and show that, for large k, one can take c(k) = O(1/k log(2k)). We also give a lower bound for the size of a set A ⊆ Zp such that every element of A+A has at least K representations in the form a + a′, a, a′ ∈ A.