Degree of independence of numbers

A new concept of independence of real numbers, called degree independence, which contains those of linear and algebraic independences, is introduced. A sufficient criterion for such independence is established based on a 1988 result of Bundschuh, which in turn makes use of a generalization of Liouville’s estimate due to Feldman in 1968. Applications to numbers represented by Cantor series and product expansions are derived.

BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES

We introduce a numeration system, called the <em>beta Cantor series expansion</em>, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix $\gamma \in \mathbb{R}$ and a sequence $B=\{\beta_i\}$ of real number bases, every element of the interval $x \in [\gamma,\gamma+1)$ has a <em>beta Cantor series expansion</em> with respect to B where the digits are integers in some alphabet $\mathcal{A}(B)$. We give a criterion in determining whether an integer sequence is admissible when $B$ satisfies some condition. We provide a description of the reference strings, namely the expansion of $\gamma$ and $\gamma+1$, used in the admissibility criterion.