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is satisfied by infinitely many rational fractions p/q when fe^l/S, and further, that there exist irrationals everywhere dense on the real axis for which (1) is satisfied by only a finite number of fractions when k < 1/S. He used simple continued fractions to get this result. The same result has since been obtained in two different ways by Ford. If o denotes an odd integer and e an even integer, then all irreducible fractions p/q are of three classes [o/e], [e/o], and [0/0]. It will be shown that If k ^ 1, there are infinitely many fractions of each of the three classes satisfying (1), regardless of the value of the real irrational number o). If k<l, there exist irrational numbers everywhere dense on the real axis f or which (1) is satisfied by only a finite number of f raclions of a given one of the three classes. The proof, like Ford's first proof of Hurwitz' theorem, will depend to a large extent on geometric properties of elliptic modular transformations,

Approximation to real irrationals by certain classes of rational fractions