EXISTENCE AND OPTIMALITY OF w-NON-ADJACENT FORMS WITH AN ALGEBRAIC INTEGER BASE

We consider digital expansions in lattices with endomorphisms acting as base. We focus on the w-non-adjacent form (w-NAF), where each block of w consecutive digits contains at most one non-zero digit. We prove that for su ciently large w and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a w-NAF. If the eigenvalues of the endomorphism are large enough and w is su ciently large, then the w-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.