Abstract:
We prove a new transference theorem in the geometry of numbers, giving optimal bounds relating the successive minima of a lattice with the minimal length of generating vectors of its dual. It generalizes the transference theorem due to Banaszczyk. We also prove a stronger bound for the special class of lattices possessing $n^{\epsilon}$-unique shortest lattice vectors. The theorems imply consequent improvement of the Ajtai connection factors in the connection of average-case to worst-case complexity of the shortest lattice vector problem. Our proofs are non-constructive, based on methods from harmonic analysis.