The authors prove the following theorem: For every \(n \geq 1\) there exist \(2^{\aleph_0}\) many \(q \in(1,2)\) such that 1 has exactly \(n+1\) expansions of the form \(1 = \sum^\infty_{i=1} \varepsilon_i/q^i\), where the digits \(\varepsilon_i\) can be 0 or 1. [For part II see the review below].