On a system of step functions (Q1608587)

In connection with B. Nyman's approach to the Riemann hypothesis the following problem is considered: describe all finite sums of the form \((*)\) \(\sum c_k[x/m_k]\), \(m_k\in \mathbb{N}\), taking values in \([0,1]\); here \([t]\) denotes the integer part of \(t\). Some special cases are investigated, where the sum in \((*)\) is finite, and some conjectures are formulated.