Statistic and ergodic properties of Minkowski's diagonal continued fraction (Q1119683)

Every irrational x can be represented in a unique way as a semi-regular continued fraction whose convergents are exactly the reduced fractions p/q with \(\theta:=q | qx-p| <1/2.\) This expansion is Minkowski's diagonal continued fraction. In earlier works the author has introduced the concept of S-expansions. It is shown that Minkowski's diagonal continued fraction is a special example of an S-expansion, and supported by knowing this fact, some metrical results on distribution properties of some sequences connected with Minkowski's diagonal expansion are shown. As an example: Let x be an irrational number and let \((a_ k)_{k\geq 0}\) be the sequence of the diagonal continued fraction-partial quotients of x. Then \[ \lim_{N\to \infty}(1/N)\quad \#\{j\leq N;\quad a_ j=1\}=2(\log (\sqrt{2}-1)+\sqrt{2}-1/2)=0.0656... \] almost everywhere. Finally an algorithm is given to gain Minkowski's diagonal continued fraction expansion of x from the regular continued fraction expansion of x.