A new class of continued fraction expansions

Maximal $S$-expansions are Bernoulli shifts

“…which needs some minor modifications in order to satisfy the above definition 1, see *9* from remark 1, yields the nearest integer continued fraction (NICF). The area 6nicf is maximal; see also [Kl,sect. 4…”

“…Put 5i := S^S*, 62 :== S\S^ (see also figure 1). Then by invariance of fi, due to definition 1 (iii) and from the fact that S^ U T6'* covers the rectangle containing any singularization area it now follows that : see also [Kl,thm (4.7)]. A singularization area is called maximal in casê…”

“…where ( We have the following corollary of the proof of THEOREM 5, see also [Kl,thm (4.11 One could wonder whether the converse of this corollary holds; i.e. doeŝ i(Bs) = 0 («with probability 1 no partial quotient equal to 1 survives))) imply that S is maximal ?…”

“…(ii) It is impossible to singularize in the RCF-expansion (1) of an irrational number x a partial quotient greater than 1, and still obtain a SRCF which converges to x, (see [Kl,cor. 1.10…”

“…-In [Kl,sect. 4] (for the NICF) and in [BK2] (for the OCF) it is shown, that in order to obtain the NICF (resp.…”