THE CONTINUOUS DIOPHANTINE APPROXIMATION MAPPING OF SZEKERES

Szekeres defined a continuous analogue of the additive ordinary contin ued fraction expansion, which iterates a map T on a domain which can b e identified with the unit square [0, 1](2). Associated to it are cont inuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spect ra, respectively; thus providing an alternative characterization of th em. Szekeres also described a multi-dimensional analogue of T, which i terates a map T-d on a higher dimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formul ate several open problems concerning the Diophantine approximation pro perties of the map T-d.