On the distance between Frobenius numbers

Let n ≥ 2 and k ≥ 1 be integers and a = (a_1,...,a_n) be an integer vector with positive coprime entries. The k-Frobenius number F_k(a) is the largest integer that cannot be represented as a nonnegative integer combination of a_i in at least k different ways. We study the quantity (F_k(a) − F_1(a))(a1···an)^(−1/(n−1)) and use obtained results to improve existing upper bounds for 2-Frobenius numbers. The proofs are based on packing and covering results from the geometry of numbers