Digital representations using the greatest integer function

Let S_d(α) denote the set of all integers which can be expressed in the form Σ ε[αⁱ] with ε_i € (0,…, d ™ 1), where d ≥ 2 is an integer and α ≥ 1 is real, and let l_d denote the set of a so that S_d(α) = Z⁺. We show that I_d = [1, rd]∪(d), where r2 = 13^l/4, r₃ = 22^1/3 and r₄ = (d² − d − 2)^l/2 for d ≥ 4. If α Ε I_d, we show that T_d(α), the complement of S_d(α), is infinite, and discuss the density of T_d(α) when α < d. For d ≥ 4 and a particular quadratic irrational β = β(d) < d, we describe T_d(β) explicitly and show that ΙT_d(β) ∩ [0, n]Ι is of order n^e(d), where e(d) > 1.