A parameterized family of weighted Ducci maps

A simple type of unbounded, zero-sum weighting, defined by a single parameter, is applied to the Ducci map on [image omitted]. It is shown how varying this parameter affects the dynamics of the corresponding Ducci sequences, and a connection between the dynamics and the golden ratio is established. These results also establish the existence of unbounded, zero-sum Ducci map weightings for which only the zero-vector is a cycle. Finally, by considering Roth's theorem, it is shown how the algebraic irrational numbers are related to the Ducci problem on [image omitted].