The Joint Distribution of Greedy and Lazy Fibonacci Expansions

with digits k ∈ {0, 1}. The maximal expansion with respect to the lexicographic order on (. . . , 4, 3, 2) is the Zeckendorf expansion or, more generally, the greedy expansion, which has been studied by Zeckendorf [7] and many others. (Lexicographic order means (. . . , 3, 2) < (. . . , 3, ′ 2) if k < ′ k for some k ≥ 2 and j ≤ j for all j ≥ k.) The minimal expansion with respect to this order is the less known lazy expansion, which was introduced by Erdős and Joó [4] (for q-ary expansions of 1, 1 < q < 2). For example, 100 has greedy expansion 100 = 89 + 8 + 3 = F11 + F6 + F4 and lazy expansion 100 = 55 + 21 + 13 + 5 + 3 + 2 + 1 = F10 + F8 + F7 + F5 + F4 + F3 + F2. Denote the digits of the greedy expansion by g k(n) and those of the lazy expansion by k(n). The aim of this work is to study the structure of the possible digit sequences in order to obtain distributional results for the sum-of-digits functions sg(n) = ∑