A Cantor-like SetConsider the collection {$\mathcal T$} of all infinite continued tractions which do not include the traction, {$C$}. There are uncountably many of these. If {$T$} is one such, {$F_0F_1F_2\ \ldots$}, it is easy to see that {$$\Delta^k T=F_kF_{k+1}F_{k+2}\ \ldots$$} is never strictly triangular. It may, however be a rim. The infinite continued traction, {$MLLL\overline{L}\ \ldots$}, converges to {$\langle .5,.5,0)$}. {$MLRLR\overline{LR}\ \ldots$}, converges to {$\langle .25,.5,.25)$}. We proved (There are abnormal boxes) that there are abnormal boxes. The analysis of continued tractions shows that there are uncountably many of them. This set is also closed---it's clear that if a sequence of continued tractions from {$\mathcal T$} converges, it converges to a member of {$\mathcal T$}. The set of numbers representable by members of {$\mathcal T$} is the Rauzy gasket and is the union of the abnormal boxes and rims.  In "The Rauzy Gasket" Arnoux and Starosta show that the Rauzy gasket has Lebesgue measure 0. This gives us that almost every box is normal. |