A two-dimensional singular function via Sturmian words in base β

For β>1, a function (⋅)β maps each infinite word a1a2⋯∈NN to a real number ∑i=1∞ai/βi. We define Ξ(α,β) by (sα,0′)β where sα,0′ is a lexicographically greatest mechanical word of slope α. This paper demonstrates that the function Ξ enjoys devilʼs staircase-like properties. Its continuity, partial and total differentiability will be investigated. We also present a set of Ξ-values, in which any finite number of members are algebraically independent over the field of rationals.