Multiple Kakeya expansions

We are interested in expansions of the form $x=\sum c_nt_n$ with digits $c_i$ of zeros and ones, where $(t_n)$ is a given sequence of positive real numbers. Kakeya gave a classical theorem ensuring that under a natural condition on the sequence every $x\in[0,\sum t_n]$ has at least one expansion. We give two stronger conditions ensuring that every $x\in(0,\sum t_n)$ has $2^{\aleph_0}$ expansions. The first one leads to significantly shorter proofs of the existence of $2^{\aleph_0}$ expansions where $(1/t_n)$ is a certain Fibonacci or Lucas type sequence, recently proved by a quasi-ergodic approach. The second one allows us to prove analogous results for non-integer base expansions on arbitrary ternary alphabets.