Pisot numbers and strong negations

Abstract We introduce and study representation systems for the numbers in the unit interval [0, 1]. We call them ϕ m -systems (where ϕ m is a pseudo-golden ratio). With the aid of these representation systems, we define a family h m of strong negations and an increasing function g m which is the inverse of the generator of h m . The functions h m and g m are singular, and we study several properties; among which we calculate the Hausdorff dimensions of certain sets that are related to them. Finally, we prove that g m is an infinite convolution, and the sequence of coefficients in the Fourier series of its associated Stieltjes measure does not converge to zero.