Existence and Optimality of $w$-Non-adjacent Forms with an Algebraic Integer Base

TL;DR

This paper proves the existence and optimality of $w$-non-adjacent forms in lattice digital expansions with algebraic integer bases, showing they minimize weight under certain conditions.

Contribution

It establishes conditions for the existence and optimality of $w$-NAF expansions in lattice bases with algebraic integers, extending previous results to more general settings.

Findings

Existence of $w$-NAF for large $w$ and expanding endomorphisms.

$w$-NAF minimizes weight among all expansions under certain eigenvalue conditions.

Results apply to digital expansions in algebraic lattice bases.

Abstract

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 sufficiently 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 sufficiently 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.