The level of distribution of the sum-of-digits function of linear recurrence number systems

Let G=(G j ) j≥0 be a strictly increasing linear recurrent sequence of integers with G 0 =1 having characteristic polynomial X d -a 1 d-1 -⋯-a X-a . It is well known that each positive integer ν can uniquely represented by the so-called greedy expansion ν=ε (ν)G +⋯+ε ℓ for ℓ∈ℕ satisfying ≤ν0