Frobenius Problem and the Covering Radius of a Lattice

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as P N i=1 aixi where x1, ..., xN are non-negative integers. The condition that gcd(a1 , ..., aN ) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1, ..., aN is NP-hard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.