For any pair $i,j\ge 0$ with $i+j=1$ let ${\mathbf Bad}(i,j)$ denote the set of pairs $(\alpha,\beta)\in {\bb R}^2$ for which $\max\{\|q\alpha\|^{1/i}\|q\beta\|^{1/j}\}>c/q$ for all $q\in {\bb N}$ . Here $c=c(\alpha,\beta)$ is a positive constant. If $i=0$ the set ${\mathbf Bad}(0, 1)$ is identified with ${\bb R}\times {\mathbf Bad}$ where ${\mathbf Bad}$ is the set of badly approximable numbers. That is, ${\mathbf Bad}(0, 1)$ consists of pairs $(\alpha, \beta)$ with $\alpha\in {\bb R}$ and $\beta\in {\mathbf Bad}$ If $j=0$ the roles of $\alpha$ and $\beta$ are reversed. It is proved that the set ${\mathbf Bad}(1,0)\cap {\mathbf Bad} (0,1)\cap {\mathbf Bad}(i,j)$ has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.

Szekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.

Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

It is shown that there exists a sequence of integers a1 < a2 <... such that each positive integer is a difference of elements of the sequence in exactly one way, and such that ak does not exceed a constant times k3. In fact we construct such a sequence with each ak in [C(k -1)3, Ck3), where C is an absolute constant.