Continuum cardinality of the set of solutions of one class of equations that contain the function of frequency of ternary digits of a number

Abstract

We study the equation v₁ (x) = x, where v₁ (x) is the function of frequency of the digit 1 in ternary expansion of x. We prove that this equation has a unique rational solution and a continuum set of irrational solutions. An algorithm for the construction of solutions is proposed. We also describe the topological and metric properties of the set of all solutions. Some additional facts about equations v_i (x), i = 0,2, are also given.