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

Let G = (Gj )j≥0 be a strictly increasing linear recurrent sequence of integers with G0 = 1 having characteristic polynomial Xd − a1Xd−1 − · · · − ad−1X − ad. It is well known that each positive integer ν can be uniquely represented by the so-called greedy expansion ν = ε0(ν)G0 + · · · + εℓ(ν)Gℓ for ℓ ∈ N satisfying Gℓ ≤ ν < Gℓ+1. Here the digits are defined recursively in a way that 0 ≤ ν − εℓ(ν)Gℓ − · · · − εj (ν)Gj < Gj holds for 0 ≤ j ≤ ℓ. In the present paper we study the sum-of-digits function sG(ν) = ε0(ν) + · · · + εℓ(ν) under certain natural assumptions on the sequence G. In particular, we determine its level of distribution x ϑ. ...