Frobenius Problem and the Covering Radius of a Lattice

Let $N \geq2$ and let $1 < a_1 < \cdots < a_N$ be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as $\sum_{i=1}^N a_i x_i {\rm where\ } x_1,\ldots,x_N$ are non-negative integers. The condition that $\gcd(a_1,\ldots,a_N)=1$ implies that such a number exists. The general problem of determining the Frobenius number given N and $a_1,\ldots,a_N$ is NP-hard, but there have 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 the 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.